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Properties

Label 1620.2.d.d
Level $1620$
Weight $2$
Character orbit 1620.d
Analytic conductor $12.936$
Analytic rank $0$
Dimension $6$
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] = [1620,2,Mod(649,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.d (of order \(2\), degree \(1\), 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: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.301925376.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 14x^{4} + 43x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{7} + \beta_{2} q^{11} + ( - \beta_{5} - \beta_1) q^{13} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{17} + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{19} + (\beta_{4} - \beta_{3} - 3 \beta_1) q^{23} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{25}+ \cdots + ( - 4 \beta_{5} + \beta_{4} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{5} + 2 q^{11} + 3 q^{25} - 18 q^{29} - 6 q^{31} + 17 q^{35} + 14 q^{41} - 3 q^{55} - 34 q^{59} - 6 q^{61} + 15 q^{65} + 6 q^{79} + 12 q^{85} - 56 q^{89} + 6 q^{91} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 14x^{4} + 43x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 6\nu^{4} + 14\nu^{3} + 72\nu^{2} + 31\nu + 114 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 6\nu^{4} - 14\nu^{3} + 72\nu^{2} - 31\nu + 114 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 12\nu^{3} - 19\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 3\beta_{4} + 3\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} - 12\beta_{2} + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{5} + 36\beta_{4} - 36\beta_{3} + 53\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.20590i
− 1.20590i
− 3.17695i
3.17695i
1.56613i
− 1.56613i
0 0 0 −1.83216 − 1.28187i 0 3.76963i 0 0 0
649.2 0 0 0 −1.83216 + 1.28187i 0 − 3.76963i 0 0 0
649.3 0 0 0 0.123563 − 2.23265i 0 1.28835i 0 0 0
649.4 0 0 0 0.123563 + 2.23265i 0 − 1.28835i 0 0 0
649.5 0 0 0 2.20860 − 0.349411i 0 2.26496i 0 0 0
649.6 0 0 0 2.20860 + 0.349411i 0 − 2.26496i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.d.d 6
3.b odd 2 1 1620.2.d.c 6
5.b even 2 1 inner 1620.2.d.d 6
5.c odd 4 2 8100.2.a.bd 6
9.c even 3 2 540.2.r.a 12
9.d odd 6 2 180.2.r.a ✓ 12
15.d odd 2 1 1620.2.d.c 6
15.e even 4 2 8100.2.a.bc 6
36.f odd 6 2 2160.2.by.e 12
36.h even 6 2 720.2.by.e 12
45.h odd 6 2 180.2.r.a ✓ 12
45.j even 6 2 540.2.r.a 12
45.k odd 12 4 2700.2.i.f 12
45.l even 12 4 900.2.i.f 12
180.n even 6 2 720.2.by.e 12
180.p odd 6 2 2160.2.by.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.r.a ✓ 12 9.d odd 6 2
180.2.r.a ✓ 12 45.h odd 6 2
540.2.r.a 12 9.c even 3 2
540.2.r.a 12 45.j even 6 2
720.2.by.e 12 36.h even 6 2
720.2.by.e 12 180.n even 6 2
900.2.i.f 12 45.l even 12 4
1620.2.d.c 6 3.b odd 2 1
1620.2.d.c 6 15.d odd 2 1
1620.2.d.d 6 1.a even 1 1 trivial
1620.2.d.d 6 5.b even 2 1 inner
2160.2.by.e 12 36.f odd 6 2
2160.2.by.e 12 180.p odd 6 2
2700.2.i.f 12 45.k odd 12 4
8100.2.a.bc 6 15.e even 4 2
8100.2.a.bd 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{6} + 21T_{7}^{4} + 105T_{7}^{2} + 121 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 22T_{11} + 46 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 21 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 22 T + 46)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 39 T^{4} + \cdots + 324 \) Copy content Toggle raw display
$17$ \( T^{6} + 36 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 42 T + 72)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 93 T^{4} + \cdots + 28561 \) Copy content Toggle raw display
$29$ \( (T + 3)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} + 3 T^{2} + \cdots - 358)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 60 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( (T^{3} - 7 T^{2} - 19 T + 97)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 231 T^{4} + \cdots + 91204 \) Copy content Toggle raw display
$47$ \( T^{6} + 105 T^{4} + \cdots + 961 \) Copy content Toggle raw display
$53$ \( T^{6} + 264 T^{4} + \cdots + 331776 \) Copy content Toggle raw display
$59$ \( (T^{3} + 17 T^{2} + \cdots + 88)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 3 T^{2} - 39 T + 31)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 105 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$71$ \( (T^{3} - 42 T + 72)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 384 T^{4} + \cdots + 369664 \) Copy content Toggle raw display
$79$ \( (T^{3} - 3 T^{2} + \cdots + 108)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 405 T^{4} + \cdots + 1907161 \) Copy content Toggle raw display
$89$ \( (T^{3} + 28 T^{2} + \cdots + 662)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 339 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
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