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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + (\beta_{3} + \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{3} - \beta_{2} - 2) q^{6} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{2} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + (\beta_{3} + \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{3} - \beta_{2} - 2) q^{6} + (\beta_{3} + \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + (\beta_{3} - \beta_{2}) q^{10} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_1 + 2) q^{12} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 2) q^{13} + (\beta_{3} - 3 \beta_{2} - 2) q^{14} + ( - \beta_{3} + \beta_1 - 1) q^{15} + ( - 2 \beta_{3} + 2 \beta_1) q^{16} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 2) q^{17} + 6 q^{19} + ( - \beta_{3} + \beta_1 - 2) q^{20} + ( - 2 \beta_{3} + 2 \beta_1 + 1) q^{21} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{22} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{23} + (2 \beta_{2} - 4 \beta_1 + 2) q^{24} - q^{25} + ( - 3 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 4) q^{26} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{27} + ( - 3 \beta_{3} + 3 \beta_1 - 2) q^{28} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 1) q^{29} + ( - \beta_{2} - \beta_1 + 2) q^{30} - 6 q^{31} + ( - 4 \beta_1 + 4) q^{32} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{33} + (3 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{34} + ( - \beta_{3} + \beta_1 - 3) q^{35} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{37} + (6 \beta_{2} - 6 \beta_1) q^{38} + (3 \beta_{3} - 6 \beta_{2} - 3 \beta_1 + 3) q^{39} + ( - 2 \beta_{2} + 2) q^{40} + (2 \beta_{3} - 2 \beta_1 + 2) q^{41} + (\beta_{2} - 5 \beta_1 + 4) q^{42} + 2 \beta_{2} q^{43} + ( - 3 \beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{44} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{46} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{47} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{48} + ( - 4 \beta_{3} + 4 \beta_1 - 5) q^{49} + ( - \beta_{2} + \beta_1) q^{50} + ( - 3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 3) q^{51} + (5 \beta_{3} - 8 \beta_{2} - \beta_1 + 6) q^{52} + 2 q^{53} + (3 \beta_{3} + 3 \beta_{2} + 6) q^{54} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{55} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{56} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{57} + (4 \beta_{3} + 5 \beta_{2} - \beta_1 + 8) q^{58} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{59} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{60} + ( - 2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 2) q^{61} + ( - 6 \beta_{2} + 6 \beta_1) q^{62} + 8 \beta_{2} q^{64} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{65} + (3 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 4) q^{66} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{67} + ( - \beta_{3} - 8 \beta_{2} + 5 \beta_1 - 6) q^{68} + (2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 2) q^{69} + ( - 3 \beta_{2} + \beta_1 + 2) q^{70} + (2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{71} + ( - 4 \beta_{3} - 6 \beta_{2} + 4 \beta_1 - 4) q^{73} + ( - 2 \beta_{3} - 8 \beta_{2} + 6 \beta_1 - 4) q^{74} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{75} + (6 \beta_{3} + 6 \beta_1) q^{76} + ( - 4 \beta_{3} + 5 \beta_{2} - 6) q^{77} + ( - 6 \beta_{3} + 9 \beta_{2} + 3 \beta_1 - 6) q^{78} + (5 \beta_{3} + 6 \beta_{2} - 5 \beta_1 + 5) q^{79} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{80} - 9 q^{81} + (2 \beta_{2} + 2 \beta_1 - 4) q^{82} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 12) q^{83} + (\beta_{3} + 8 \beta_{2} + \beta_1) q^{84} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{85} + (2 \beta_{3} - 2 \beta_{2}) q^{86} + (\beta_{3} - \beta_{2} + \beta_1 - 12) q^{87} + (4 \beta_{3} - 6 \beta_{2} + 6) q^{88} + (2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 2) q^{89} + (7 \beta_{3} - 10 \beta_{2} + \beta_1 + 9) q^{91} + (4 \beta_{3} - 8 \beta_{2} + 4) q^{92} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1) q^{93} + (\beta_{3} + \beta_{2} + 2) q^{94} + 6 \beta_{2} q^{95} + (4 \beta_{3} - 8 \beta_{2} - 4) q^{96} + (6 \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 6) q^{97} + ( - 5 \beta_{2} - 3 \beta_1 + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 6 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 6 q^{6} - 8 q^{8} - 2 q^{10} + 12 q^{12} - 10 q^{14} + 8 q^{16} + 24 q^{19} - 4 q^{20} + 12 q^{21} + 2 q^{22} - 4 q^{25} - 6 q^{26} + 4 q^{28} + 4 q^{29} + 6 q^{30} - 24 q^{31} + 8 q^{32} - 18 q^{34} - 8 q^{35} - 24 q^{37} - 12 q^{38} + 8 q^{40} + 6 q^{42} - 8 q^{44} - 8 q^{46} - 4 q^{49} + 2 q^{50} + 12 q^{52} + 8 q^{53} + 18 q^{54} - 8 q^{55} + 16 q^{56} + 22 q^{58} + 12 q^{62} + 12 q^{65} + 6 q^{66} - 12 q^{68} + 10 q^{70} - 16 q^{77} - 6 q^{78} - 36 q^{81} - 12 q^{82} + 48 q^{83} - 12 q^{85} - 4 q^{86} - 48 q^{87} + 16 q^{88} + 24 q^{91} + 8 q^{92} + 6 q^{94} - 24 q^{96} + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(140, [\chi])\):

\( T_{3}^{2} - 3 \) Copy content Toggle raw display
\( T_{19} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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