↑

Properties

Label 420.2.w.b
Level $420$
Weight $2$
Character orbit 420.w
Analytic conductor $3.354$
Analytic rank $0$
Dimension $8$
CM discriminant -84
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [420,2,Mod(83,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 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("420.83");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.w (of order \(4\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (\beta_{7} - \beta_{3}) q^{5} + (\beta_{7} + \beta_1) q^{6} + (\beta_{7} + \beta_{4} - \beta_{3} - \beta_1) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (\beta_{7} - \beta_{3}) q^{5} + (\beta_{7} + \beta_1) q^{6} + (\beta_{7} + \beta_{4} - \beta_{3} - \beta_1) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + 3 \beta_{2} q^{9} + (\beta_{7} - \beta_{4} - \beta_{3}) q^{10} + ( - \beta_{6} + \beta_{5} - 2) q^{11} + 2 \beta_{7} q^{12} + (\beta_{7} - 2 \beta_{3} - \beta_1) q^{14} + (\beta_{6} + 2 \beta_{2} + 2) q^{15} - 4 q^{16} + (2 \beta_{7} - \beta_{4} + \cdots + \beta_1) q^{17}+ \cdots + (3 \beta_{6} + 3 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 16 q^{8} - 8 q^{11} + 12 q^{15} - 32 q^{16} + 24 q^{18} - 8 q^{22} - 16 q^{25} + 24 q^{30} - 32 q^{32} - 28 q^{35} + 48 q^{36} + 32 q^{37} - 16 q^{50} + 72 q^{51} - 48 q^{57} + 24 q^{60} - 56 q^{70} - 88 q^{71} + 48 q^{72} - 72 q^{81} + 64 q^{85} + 16 q^{88} - 24 q^{93} + 56 q^{95} - 56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 23x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 19\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 24\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 24\nu^{3} + 5\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 24\nu^{3} + 24\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - \nu^{4} - 67\nu^{2} - 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + \nu^{4} - 67\nu^{2} + 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{7} - 91\nu^{3} ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + 6\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{4} + 2\beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{6} - 5\beta_{5} - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{4} - 19\beta_{3} + 29\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{6} - 12\beta_{5} - 67\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -48\beta_{7} + 91\beta_{4} - 91\beta_{3} - 91\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
1.54779 + 1.54779i
−0.323042 − 0.323042i
0.323042 + 0.323042i
−1.54779 − 1.54779i
−0.323042 + 0.323042i
1.54779 − 1.54779i
−1.54779 + 1.54779i
0.323042 − 0.323042i
1.00000 − 1.00000i −1.22474 − 1.22474i − 2.00000i −1.22474 − 1.87083i −2.44949 1.87083 − 1.87083i −2.00000 − 2.00000i 3.00000i −3.09557 − 0.646084i
83.2 1.00000 − 1.00000i −1.22474 − 1.22474i − 2.00000i −1.22474 + 1.87083i −2.44949 −1.87083 + 1.87083i −2.00000 − 2.00000i 3.00000i 0.646084 + 3.09557i
83.3 1.00000 − 1.00000i 1.22474 + 1.22474i − 2.00000i 1.22474 − 1.87083i 2.44949 1.87083 − 1.87083i −2.00000 − 2.00000i 3.00000i −0.646084 − 3.09557i
83.4 1.00000 − 1.00000i 1.22474 + 1.22474i − 2.00000i 1.22474 + 1.87083i 2.44949 −1.87083 + 1.87083i −2.00000 − 2.00000i 3.00000i 3.09557 + 0.646084i
167.1 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −1.22474 − 1.87083i −2.44949 −1.87083 − 1.87083i −2.00000 + 2.00000i − 3.00000i 0.646084 − 3.09557i
167.2 1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i −1.22474 + 1.87083i −2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i − 3.00000i −3.09557 + 0.646084i
167.3 1.00000 + 1.00000i 1.22474 − 1.22474i 2.00000i 1.22474 − 1.87083i 2.44949 −1.87083 − 1.87083i −2.00000 + 2.00000i − 3.00000i 3.09557 − 0.646084i
167.4 1.00000 + 1.00000i 1.22474 − 1.22474i 2.00000i 1.22474 + 1.87083i 2.44949 1.87083 + 1.87083i −2.00000 + 2.00000i − 3.00000i −0.646084 + 3.09557i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
5.c odd 4 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
35.f even 4 1 inner
60.l odd 4 1 inner
420.w even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.w.b yes 8
3.b odd 2 1 420.2.w.a ✓ 8
4.b odd 2 1 420.2.w.a ✓ 8
5.c odd 4 1 inner 420.2.w.b yes 8
7.b odd 2 1 inner 420.2.w.b yes 8
12.b even 2 1 inner 420.2.w.b yes 8
15.e even 4 1 420.2.w.a ✓ 8
20.e even 4 1 420.2.w.a ✓ 8
21.c even 2 1 420.2.w.a ✓ 8
28.d even 2 1 420.2.w.a ✓ 8
35.f even 4 1 inner 420.2.w.b yes 8
60.l odd 4 1 inner 420.2.w.b yes 8
84.h odd 2 1 CM 420.2.w.b yes 8
105.k odd 4 1 420.2.w.a ✓ 8
140.j odd 4 1 420.2.w.a ✓ 8
420.w even 4 1 inner 420.2.w.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.w.a ✓ 8 3.b odd 2 1
420.2.w.a ✓ 8 4.b odd 2 1
420.2.w.a ✓ 8 15.e even 4 1
420.2.w.a ✓ 8 20.e even 4 1
420.2.w.a ✓ 8 21.c even 2 1
420.2.w.a ✓ 8 28.d even 2 1
420.2.w.a ✓ 8 105.k odd 4 1
420.2.w.a ✓ 8 140.j odd 4 1
420.2.w.b yes 8 1.a even 1 1 trivial
420.2.w.b yes 8 5.c odd 4 1 inner
420.2.w.b yes 8 7.b odd 2 1 inner
420.2.w.b yes 8 12.b even 2 1 inner
420.2.w.b yes 8 35.f even 4 1 inner
420.2.w.b yes 8 60.l odd 4 1 inner
420.2.w.b yes 8 84.h odd 2 1 CM
420.2.w.b yes 8 420.w even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 2T_{11} - 20 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 20)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 3824 T^{4} + 160000 \) Copy content Toggle raw display
$19$ \( (T^{4} + 76 T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1764)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 124 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 14)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 22 T + 100)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
show more
show less