Properties

Label 140.2.w.a
Level $140$
Weight $2$
Character orbit 140.w
Analytic conductor $1.118$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [140,2,Mod(23,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.w (of order \(12\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.12745506816.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 49x^{4} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2} - 1) q^{2} + \beta_1 q^{3} + (2 \beta_{6} - 2 \beta_{2}) q^{4} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{5} + (\beta_{5} + \beta_{3} - \beta_1) q^{6} + \beta_{7} q^{7} + ( - 2 \beta_{6} - 2) q^{8} + 4 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2} - 1) q^{2} + \beta_1 q^{3} + (2 \beta_{6} - 2 \beta_{2}) q^{4} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{5} + (\beta_{5} + \beta_{3} - \beta_1) q^{6} + \beta_{7} q^{7} + ( - 2 \beta_{6} - 2) q^{8} + 4 \beta_{2} q^{9} + ( - 3 \beta_{6} - \beta_{4} + 3 \beta_{2}) q^{10} + ( - \beta_{7} + \beta_1) q^{11} + (2 \beta_{7} - 2 \beta_{3}) q^{12} + ( - 2 \beta_{6} - 2) q^{13} + (\beta_{5} - \beta_{3} - \beta_1) q^{14} + ( - \beta_{5} - 2 \beta_{3} + \beta_1) q^{15} + ( - 4 \beta_{4} + 4) q^{16} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{17} + (4 \beta_{6} + 4 \beta_{4} - 4 \beta_{2}) q^{18} + ( - \beta_{7} - \beta_{5} + \beta_{3}) q^{19} + (2 \beta_{6} + 4) q^{20} + (7 \beta_{4} - 7) q^{21} + 2 \beta_{3} q^{22} + \beta_{5} q^{23} + ( - 2 \beta_{7} - 2 \beta_1) q^{24} + (4 \beta_{6} + 3 \beta_{4} - 4 \beta_{2}) q^{25} + ( - 4 \beta_{4} + 4) q^{26} + \beta_{3} q^{27} - 2 \beta_{5} q^{28} - 3 \beta_{6} q^{29} + ( - 3 \beta_{7} - \beta_{5} + 3 \beta_{3}) q^{30} + (2 \beta_{7} - 2 \beta_1) q^{31} + ( - 4 \beta_{6} + 4 \beta_{4} + 4 \beta_{2}) q^{32} + ( - 7 \beta_{4} + 7 \beta_{2} + 7) q^{33} - 4 \beta_{6} q^{34} + ( - 2 \beta_{5} + \beta_{3} + 2 \beta_1) q^{35} - 8 q^{36} + 2 \beta_1 q^{38} + ( - 2 \beta_{7} - 2 \beta_1) q^{39} + (6 \beta_{4} + 2 \beta_{2} - 6) q^{40} + 3 q^{41} + (7 \beta_{6} - 7 \beta_{4} - 7 \beta_{2}) q^{42} + (3 \beta_{5} - 3 \beta_1) q^{43} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{44} + ( - 4 \beta_{6} - 8 \beta_{4} + 4 \beta_{2}) q^{45} + (\beta_{7} - \beta_1) q^{46} + ( - 4 \beta_{7} + 4 \beta_{3}) q^{47} + ( - 4 \beta_{5} + 4 \beta_1) q^{48} - 7 \beta_{2} q^{49} + ( - \beta_{6} - 7) q^{50} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3}) q^{51} + ( - 4 \beta_{6} + 4 \beta_{4} + 4 \beta_{2}) q^{52} + (5 \beta_{6} + 5 \beta_{4} - 5 \beta_{2}) q^{53} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{54} + (\beta_{5} - 3 \beta_{3} - \beta_1) q^{55} + ( - 2 \beta_{7} + 2 \beta_1) q^{56} + ( - 7 \beta_{6} + 7) q^{57} + ( - 3 \beta_{4} + 3 \beta_{2} + 3) q^{58} + (\beta_{7} + \beta_1) q^{59} + (2 \beta_{7} + 4 \beta_1) q^{60} + ( - 3 \beta_{4} + 3) q^{61} - 4 \beta_{3} q^{62} + (4 \beta_{5} - 4 \beta_1) q^{63} + 8 \beta_{6} q^{64} + (6 \beta_{4} + 2 \beta_{2} - 6) q^{65} + 14 \beta_{4} q^{66} + 5 \beta_{7} q^{67} + ( - 4 \beta_{4} + 4 \beta_{2} + 4) q^{68} + 7 \beta_{6} q^{69} + ( - \beta_{7} + 3 \beta_{5} + \beta_{3}) q^{70} + (\beta_{5} + \beta_{3} - \beta_1) q^{71} + ( - 8 \beta_{4} - 8 \beta_{2} + 8) q^{72} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{2}) q^{73} + (4 \beta_{7} + 3 \beta_{5} - 4 \beta_{3}) q^{75} + (2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{76} + (7 \beta_{4} + 7 \beta_{2} - 7) q^{77} + ( - 4 \beta_{5} + 4 \beta_1) q^{78} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{79} + (8 \beta_{6} - 4 \beta_{4} - 8 \beta_{2}) q^{80} - 5 \beta_{4} q^{81} + (3 \beta_{4} + 3 \beta_{2} - 3) q^{82} + ( - \beta_{5} + \beta_1) q^{83} - 14 \beta_{6} q^{84} + (6 \beta_{6} + 2) q^{85} + (3 \beta_{7} - 3 \beta_{5} - 3 \beta_{3}) q^{86} - 3 \beta_{7} q^{87} - 4 \beta_1 q^{88} - 3 \beta_{2} q^{89} + ( - 4 \beta_{6} + 12) q^{90} + ( - 2 \beta_{7} + 2 \beta_1) q^{91} - 2 \beta_{3} q^{92} + (14 \beta_{4} - 14 \beta_{2} - 14) q^{93} + (4 \beta_{7} + 4 \beta_1) q^{94} + (\beta_{7} - 3 \beta_1) q^{95} + ( - 4 \beta_{7} + 4 \beta_{5} + 4 \beta_{3}) q^{96} + (9 \beta_{6} - 9) q^{97} + ( - 7 \beta_{6} - 7 \beta_{4} + 7 \beta_{2}) q^{98} + ( - 4 \beta_{5} + 4 \beta_{3} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 4 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 4 q^{5} - 16 q^{8} - 4 q^{10} - 16 q^{13} + 16 q^{16} - 8 q^{17} + 16 q^{18} + 32 q^{20} - 28 q^{21} + 12 q^{25} + 16 q^{26} + 16 q^{32} + 28 q^{33} - 64 q^{36} - 24 q^{40} + 24 q^{41} - 28 q^{42} - 32 q^{45} - 56 q^{50} + 16 q^{52} + 20 q^{53} + 56 q^{57} + 12 q^{58} + 12 q^{61} - 24 q^{65} + 56 q^{66} + 16 q^{68} + 32 q^{72} - 8 q^{73} - 28 q^{77} - 16 q^{80} - 20 q^{81} - 12 q^{82} + 16 q^{85} + 96 q^{90} - 56 q^{93} - 72 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 49x^{4} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 343 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 343 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 49\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 49\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 343\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 343\beta_{7} \) Copy content Toggle raw display

Character values

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

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-\beta_{6}\) \(-1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−2.55560 0.684771i
2.55560 + 0.684771i
−2.55560 + 0.684771i
2.55560 0.684771i
−0.684771 + 2.55560i
0.684771 2.55560i
−0.684771 2.55560i
0.684771 + 2.55560i
0.366025 + 1.36603i −2.55560 0.684771i −1.73205 + 1.00000i −1.23205 1.86603i 3.74166i 0.684771 2.55560i −2.00000 2.00000i 3.46410 + 2.00000i 2.09808 2.36603i
23.2 0.366025 + 1.36603i 2.55560 + 0.684771i −1.73205 + 1.00000i −1.23205 1.86603i 3.74166i −0.684771 + 2.55560i −2.00000 2.00000i 3.46410 + 2.00000i 2.09808 2.36603i
67.1 0.366025 1.36603i −2.55560 + 0.684771i −1.73205 1.00000i −1.23205 + 1.86603i 3.74166i 0.684771 + 2.55560i −2.00000 + 2.00000i 3.46410 2.00000i 2.09808 + 2.36603i
67.2 0.366025 1.36603i 2.55560 0.684771i −1.73205 1.00000i −1.23205 + 1.86603i 3.74166i −0.684771 2.55560i −2.00000 + 2.00000i 3.46410 2.00000i 2.09808 + 2.36603i
107.1 −1.36603 + 0.366025i −0.684771 + 2.55560i 1.73205 1.00000i 2.23205 + 0.133975i 3.74166i 2.55560 + 0.684771i −2.00000 + 2.00000i −3.46410 2.00000i −3.09808 + 0.633975i
107.2 −1.36603 + 0.366025i 0.684771 2.55560i 1.73205 1.00000i 2.23205 + 0.133975i 3.74166i −2.55560 0.684771i −2.00000 + 2.00000i −3.46410 2.00000i −3.09808 + 0.633975i
123.1 −1.36603 0.366025i −0.684771 2.55560i 1.73205 + 1.00000i 2.23205 0.133975i 3.74166i 2.55560 0.684771i −2.00000 2.00000i −3.46410 + 2.00000i −3.09808 0.633975i
123.2 −1.36603 0.366025i 0.684771 + 2.55560i 1.73205 + 1.00000i 2.23205 0.133975i 3.74166i −2.55560 + 0.684771i −2.00000 2.00000i −3.46410 + 2.00000i −3.09808 0.633975i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
20.e even 4 1 inner
28.g odd 6 1 inner
35.l odd 12 1 inner
140.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.w.a 8
4.b odd 2 1 inner 140.2.w.a 8
5.b even 2 1 700.2.be.c 8
5.c odd 4 1 inner 140.2.w.a 8
5.c odd 4 1 700.2.be.c 8
7.b odd 2 1 980.2.x.f 8
7.c even 3 1 inner 140.2.w.a 8
7.c even 3 1 980.2.k.e 4
7.d odd 6 1 980.2.k.g 4
7.d odd 6 1 980.2.x.f 8
20.d odd 2 1 700.2.be.c 8
20.e even 4 1 inner 140.2.w.a 8
20.e even 4 1 700.2.be.c 8
28.d even 2 1 980.2.x.f 8
28.f even 6 1 980.2.k.g 4
28.f even 6 1 980.2.x.f 8
28.g odd 6 1 inner 140.2.w.a 8
28.g odd 6 1 980.2.k.e 4
35.f even 4 1 980.2.x.f 8
35.j even 6 1 700.2.be.c 8
35.k even 12 1 980.2.k.g 4
35.k even 12 1 980.2.x.f 8
35.l odd 12 1 inner 140.2.w.a 8
35.l odd 12 1 700.2.be.c 8
35.l odd 12 1 980.2.k.e 4
140.j odd 4 1 980.2.x.f 8
140.p odd 6 1 700.2.be.c 8
140.w even 12 1 inner 140.2.w.a 8
140.w even 12 1 700.2.be.c 8
140.w even 12 1 980.2.k.e 4
140.x odd 12 1 980.2.k.g 4
140.x odd 12 1 980.2.x.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.w.a 8 1.a even 1 1 trivial
140.2.w.a 8 4.b odd 2 1 inner
140.2.w.a 8 5.c odd 4 1 inner
140.2.w.a 8 7.c even 3 1 inner
140.2.w.a 8 20.e even 4 1 inner
140.2.w.a 8 28.g odd 6 1 inner
140.2.w.a 8 35.l odd 12 1 inner
140.2.w.a 8 140.w even 12 1 inner
700.2.be.c 8 5.b even 2 1
700.2.be.c 8 5.c odd 4 1
700.2.be.c 8 20.d odd 2 1
700.2.be.c 8 20.e even 4 1
700.2.be.c 8 35.j even 6 1
700.2.be.c 8 35.l odd 12 1
700.2.be.c 8 140.p odd 6 1
700.2.be.c 8 140.w even 12 1
980.2.k.e 4 7.c even 3 1
980.2.k.e 4 28.g odd 6 1
980.2.k.e 4 35.l odd 12 1
980.2.k.e 4 140.w even 12 1
980.2.k.g 4 7.d odd 6 1
980.2.k.g 4 28.f even 6 1
980.2.k.g 4 35.k even 12 1
980.2.k.g 4 140.x odd 12 1
980.2.x.f 8 7.b odd 2 1
980.2.x.f 8 7.d odd 6 1
980.2.x.f 8 28.d even 2 1
980.2.x.f 8 28.f even 6 1
980.2.x.f 8 35.f even 4 1
980.2.x.f 8 35.k even 12 1
980.2.x.f 8 140.j odd 4 1
980.2.x.f 8 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 49T_{3}^{4} + 2401 \) acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 49T^{4} + 2401 \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{3} - T^{2} - 10 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 49T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} - 14 T^{2} + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 8)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 14 T^{2} + 196)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 49T^{4} + 2401 \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 56 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T - 3)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3969)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 12544 T^{4} + \cdots + 157351936 \) Copy content Toggle raw display
$53$ \( (T^{4} - 10 T^{3} + 50 T^{2} - 500 T + 2500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 14 T^{2} + 196)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 30625 T^{4} + \cdots + 937890625 \) Copy content Toggle raw display
$71$ \( (T^{2} + 14)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 14 T^{2} + 196)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 18 T + 162)^{4} \) Copy content Toggle raw display
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