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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [140,2,Mod(9,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.q (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 2) q^{3} + ( - \beta_{3} + \beta_1 - 1) q^{5} + ( - \beta_{2} + \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 2) q^{3} + ( - \beta_{3} + \beta_1 - 1) q^{5} + ( - \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{11} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{13} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{15} + ( - \beta_{3} + \beta_{2} - 3) q^{17} + ( - 2 \beta_{3} + \beta_1 - 1) q^{19} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{21} + (\beta_{2} - 2 \beta_1 + 3) q^{23} + (\beta_{3} - \beta_1 - 4) q^{25} + (6 \beta_{2} - 3) q^{27} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{29} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{31} + ( - 3 \beta_1 + 3) q^{33} + (6 \beta_{2} + \beta_1 - 6) q^{35} + ( - 4 \beta_{2} + \beta_1 - 5) q^{37} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{39} + ( - \beta_{3} - \beta_{2} - \beta_1 - 7) q^{41} + (\beta_{3} - 5 \beta_{2} - \beta_1 + 3) q^{43} + (2 \beta_{2} - \beta_1 + 3) q^{47} + (3 \beta_{3} + 4 \beta_{2}) q^{49} + ( - 2 \beta_{3} + 4 \beta_{2} + \cdots - 5) q^{51}+ \cdots + (8 \beta_{2} - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 2 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{11} - 3 q^{15} - 9 q^{17} - q^{19} + 12 q^{23} - 18 q^{25} - 2 q^{29} + q^{31} + 9 q^{33} - 11 q^{35} - 27 q^{37} - 6 q^{39} - 30 q^{41} + 15 q^{47} + 5 q^{49} - 9 q^{51} + 3 q^{53} + 27 q^{55} + q^{59} + 12 q^{61} + 38 q^{65} + 18 q^{67} + 24 q^{69} + 12 q^{71} + 15 q^{73} - 27 q^{75} + 9 q^{77} + 7 q^{79} + 18 q^{81} - 5 q^{85} - 3 q^{87} - 14 q^{89} + 10 q^{91} + 3 q^{93} - 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu + 5 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} + 4\beta _1 + 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(-1 + \beta_{2}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1.63746 − 1.52274i
2.13746 + 0.656712i
2.13746 − 0.656712i
−1.63746 + 1.52274i
0 1.50000 − 0.866025i 0 −0.500000 − 2.17945i 0 −1.13746 − 2.38876i 0 0 0
9.2 0 1.50000 − 0.866025i 0 −0.500000 + 2.17945i 0 2.63746 − 0.209313i 0 0 0
109.1 0 1.50000 + 0.866025i 0 −0.500000 − 2.17945i 0 2.63746 + 0.209313i 0 0 0
109.2 0 1.50000 + 0.866025i 0 −0.500000 + 2.17945i 0 −1.13746 + 2.38876i 0 0 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
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.q.b yes 4
3.b odd 2 1 1260.2.bm.b 4
4.b odd 2 1 560.2.bw.a 4
5.b even 2 1 140.2.q.a ✓ 4
5.c odd 4 2 700.2.i.f 8
7.b odd 2 1 980.2.q.b 4
7.c even 3 1 140.2.q.a ✓ 4
7.c even 3 1 980.2.e.f 4
7.d odd 6 1 980.2.e.c 4
7.d odd 6 1 980.2.q.g 4
15.d odd 2 1 1260.2.bm.a 4
20.d odd 2 1 560.2.bw.e 4
21.h odd 6 1 1260.2.bm.a 4
28.g odd 6 1 560.2.bw.e 4
35.c odd 2 1 980.2.q.g 4
35.i odd 6 1 980.2.e.c 4
35.i odd 6 1 980.2.q.b 4
35.j even 6 1 inner 140.2.q.b yes 4
35.j even 6 1 980.2.e.f 4
35.k even 12 2 4900.2.a.bf 4
35.l odd 12 2 700.2.i.f 8
35.l odd 12 2 4900.2.a.be 4
105.o odd 6 1 1260.2.bm.b 4
140.p odd 6 1 560.2.bw.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a ✓ 4 5.b even 2 1
140.2.q.a ✓ 4 7.c even 3 1
140.2.q.b yes 4 1.a even 1 1 trivial
140.2.q.b yes 4 35.j even 6 1 inner
560.2.bw.a 4 4.b odd 2 1
560.2.bw.a 4 140.p odd 6 1
560.2.bw.e 4 20.d odd 2 1
560.2.bw.e 4 28.g odd 6 1
700.2.i.f 8 5.c odd 4 2
700.2.i.f 8 35.l odd 12 2
980.2.e.c 4 7.d odd 6 1
980.2.e.c 4 35.i odd 6 1
980.2.e.f 4 7.c even 3 1
980.2.e.f 4 35.j even 6 1
980.2.q.b 4 7.b odd 2 1
980.2.q.b 4 35.i odd 6 1
980.2.q.g 4 7.d odd 6 1
980.2.q.g 4 35.c odd 2 1
1260.2.bm.a 4 15.d odd 2 1
1260.2.bm.a 4 21.h odd 6 1
1260.2.bm.b 4 3.b odd 2 1
1260.2.bm.b 4 105.o odd 6 1
4900.2.a.be 4 35.l odd 12 2
4900.2.a.bf 4 35.k even 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{4} + 44T^{2} + 256 \) Copy content Toggle raw display
$17$ \( T^{4} + 9 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} + \cdots + 196 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$29$ \( (T^{2} + T - 14)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - T^{3} + \cdots + 196 \) Copy content Toggle raw display
$37$ \( T^{4} + 27 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$41$ \( (T^{2} + 15 T + 42)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 47T^{2} + 196 \) Copy content Toggle raw display
$47$ \( T^{4} - 15 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$53$ \( T^{4} - 3 T^{3} + \cdots + 1764 \) Copy content Toggle raw display
$59$ \( T^{4} - T^{3} + \cdots + 196 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + \cdots + 441 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 48)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 15 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$79$ \( T^{4} - 7 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$83$ \( T^{4} + 87T^{2} + 1764 \) Copy content Toggle raw display
$89$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
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