Properties

Label 140.2.i.b
Level $140$
Weight $2$
Character orbit 140.i
Analytic conductor $1.118$
Analytic rank $0$
Dimension $2$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{3} + \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} - 6 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{3} + \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} - 6 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} - 6 q^{13} + 3 q^{15} + (2 \zeta_{6} - 2) q^{17} + (3 \zeta_{6} + 6) q^{21} + 9 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 9 q^{27} + 3 q^{29} + (2 \zeta_{6} - 2) q^{31} - 6 \zeta_{6} q^{33} + (2 \zeta_{6} - 3) q^{35} - 8 \zeta_{6} q^{37} + (18 \zeta_{6} - 18) q^{39} + 5 q^{41} + q^{43} + ( - 6 \zeta_{6} + 6) q^{45} - 8 \zeta_{6} q^{47} + (3 \zeta_{6} - 8) q^{49} + 6 \zeta_{6} q^{51} + (4 \zeta_{6} - 4) q^{53} + 2 q^{55} + ( - 8 \zeta_{6} + 8) q^{59} - 7 \zeta_{6} q^{61} + ( - 12 \zeta_{6} + 18) q^{63} - 6 \zeta_{6} q^{65} + ( - 3 \zeta_{6} + 3) q^{67} + 27 q^{69} + 8 q^{71} + (14 \zeta_{6} - 14) q^{73} + 3 \zeta_{6} q^{75} + (2 \zeta_{6} + 4) q^{77} - 4 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} - q^{83} - 2 q^{85} + ( - 9 \zeta_{6} + 9) q^{87} - 13 \zeta_{6} q^{89} + ( - 18 \zeta_{6} + 6) q^{91} + 6 \zeta_{6} q^{93} - 10 q^{97} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + q^{5} + q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + q^{5} + q^{7} - 6 q^{9} + 2 q^{11} - 12 q^{13} + 6 q^{15} - 2 q^{17} + 15 q^{21} + 9 q^{23} - q^{25} - 18 q^{27} + 6 q^{29} - 2 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 18 q^{39} + 10 q^{41} + 2 q^{43} + 6 q^{45} - 8 q^{47} - 13 q^{49} + 6 q^{51} - 4 q^{53} + 4 q^{55} + 8 q^{59} - 7 q^{61} + 24 q^{63} - 6 q^{65} + 3 q^{67} + 54 q^{69} + 16 q^{71} - 14 q^{73} + 3 q^{75} + 10 q^{77} - 4 q^{79} - 9 q^{81} - 2 q^{83} - 4 q^{85} + 9 q^{87} - 13 q^{89} - 6 q^{91} + 6 q^{93} - 20 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 2.59808i 0 0.500000 + 0.866025i 0 0.500000 + 2.59808i 0 −3.00000 5.19615i 0
121.1 0 1.50000 + 2.59808i 0 0.500000 0.866025i 0 0.500000 2.59808i 0 −3.00000 + 5.19615i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.i.b 2
3.b odd 2 1 1260.2.s.b 2
4.b odd 2 1 560.2.q.a 2
5.b even 2 1 700.2.i.a 2
5.c odd 4 2 700.2.r.c 4
7.b odd 2 1 980.2.i.a 2
7.c even 3 1 inner 140.2.i.b 2
7.c even 3 1 980.2.a.a 1
7.d odd 6 1 980.2.a.i 1
7.d odd 6 1 980.2.i.a 2
21.g even 6 1 8820.2.a.k 1
21.h odd 6 1 1260.2.s.b 2
21.h odd 6 1 8820.2.a.w 1
28.f even 6 1 3920.2.a.d 1
28.g odd 6 1 560.2.q.a 2
28.g odd 6 1 3920.2.a.bi 1
35.i odd 6 1 4900.2.a.a 1
35.j even 6 1 700.2.i.a 2
35.j even 6 1 4900.2.a.v 1
35.k even 12 2 4900.2.e.b 2
35.l odd 12 2 700.2.r.c 4
35.l odd 12 2 4900.2.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 1.a even 1 1 trivial
140.2.i.b 2 7.c even 3 1 inner
560.2.q.a 2 4.b odd 2 1
560.2.q.a 2 28.g odd 6 1
700.2.i.a 2 5.b even 2 1
700.2.i.a 2 35.j even 6 1
700.2.r.c 4 5.c odd 4 2
700.2.r.c 4 35.l odd 12 2
980.2.a.a 1 7.c even 3 1
980.2.a.i 1 7.d odd 6 1
980.2.i.a 2 7.b odd 2 1
980.2.i.a 2 7.d odd 6 1
1260.2.s.b 2 3.b odd 2 1
1260.2.s.b 2 21.h odd 6 1
3920.2.a.d 1 28.f even 6 1
3920.2.a.bi 1 28.g odd 6 1
4900.2.a.a 1 35.i odd 6 1
4900.2.a.v 1 35.j even 6 1
4900.2.e.b 2 35.k even 12 2
4900.2.e.c 2 35.l odd 12 2
8820.2.a.k 1 21.g even 6 1
8820.2.a.w 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$83$ \( (T + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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