Properties

Label 1620.2.x.b
Level $1620$
Weight $2$
Character orbit 1620.x
Analytic conductor $12.936$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(53,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.x (of order \(12\), degree \(4\), not 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{4} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{3}) q^{5} + ( - \beta_{4} - \beta_1) q^{7} - 2 \beta_{2} q^{11} + ( - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_1 + 3) q^{13} + ( - \beta_{7} - \beta_{3}) q^{17} - 2 \beta_{5} q^{19} + (\beta_{7} + \beta_{6} - \beta_{3} - \beta_{2}) q^{23} + 5 \beta_{4} q^{25} - 2 \beta_{6} q^{29} + ( - 4 \beta_{4} - 4) q^{31} + ( - \beta_{7} + \beta_{3}) q^{35} + ( - 3 \beta_{5} - 3) q^{37} + (4 \beta_{7} - 4 \beta_{2}) q^{41} + ( - 3 \beta_{4} + 3 \beta_1) q^{43} + ( - 3 \beta_{6} - 3 \beta_{2}) q^{47} + (5 \beta_{5} + 5 \beta_1) q^{49} + (\beta_{7} - \beta_{3}) q^{53} + 10 \beta_{5} q^{55} + ( - 4 \beta_{6} + 4 \beta_{3}) q^{59} - 6 \beta_{4} q^{61} + ( - 3 \beta_{6} - 3 \beta_{2}) q^{65} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{67} - 2 \beta_{7} q^{71} + ( - \beta_{5} + 1) q^{73} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{3} - 2 \beta_{2}) q^{77} - 6 \beta_1 q^{79} + (3 \beta_{6} - 3 \beta_{2}) q^{83} + (5 \beta_{5} - 5 \beta_{4} + 5 \beta_1 - 5) q^{85} - 2 \beta_{3} q^{89} + 6 q^{91} + (2 \beta_{7} - 2 \beta_{2}) q^{95} + (9 \beta_{4} + 9 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} + 12 q^{13} - 20 q^{25} - 16 q^{31} - 24 q^{37} + 12 q^{43} + 24 q^{61} + 4 q^{67} + 8 q^{73} - 20 q^{85} + 48 q^{91} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 13\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 29\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{6} + 24\nu^{4} - 56\nu^{2} + 21 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 22\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 3\beta_{4} - \beta_{3} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{6} + 7\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 11\beta_{5} - 5\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{2} + 29\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(\beta_{5}\) \(1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
1.40126 0.809017i
−0.535233 + 0.309017i
−1.40126 + 0.809017i
0.535233 0.309017i
−1.40126 0.809017i
0.535233 + 0.309017i
1.40126 + 0.809017i
−0.535233 0.309017i
0 0 0 −1.11803 + 1.93649i 0 1.36603 + 0.366025i 0 0 0
53.2 0 0 0 1.11803 1.93649i 0 1.36603 + 0.366025i 0 0 0
377.1 0 0 0 −1.11803 + 1.93649i 0 −0.366025 + 1.36603i 0 0 0
377.2 0 0 0 1.11803 1.93649i 0 −0.366025 + 1.36603i 0 0 0
593.1 0 0 0 −1.11803 1.93649i 0 −0.366025 1.36603i 0 0 0
593.2 0 0 0 1.11803 + 1.93649i 0 −0.366025 1.36603i 0 0 0
917.1 0 0 0 −1.11803 1.93649i 0 1.36603 0.366025i 0 0 0
917.2 0 0 0 1.11803 + 1.93649i 0 1.36603 0.366025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.x.b 8
3.b odd 2 1 inner 1620.2.x.b 8
5.c odd 4 1 inner 1620.2.x.b 8
9.c even 3 1 60.2.i.a 4
9.c even 3 1 inner 1620.2.x.b 8
9.d odd 6 1 60.2.i.a 4
9.d odd 6 1 inner 1620.2.x.b 8
15.e even 4 1 inner 1620.2.x.b 8
36.f odd 6 1 240.2.v.b 4
36.h even 6 1 240.2.v.b 4
45.h odd 6 1 300.2.i.a 4
45.j even 6 1 300.2.i.a 4
45.k odd 12 1 60.2.i.a 4
45.k odd 12 1 300.2.i.a 4
45.k odd 12 1 inner 1620.2.x.b 8
45.l even 12 1 60.2.i.a 4
45.l even 12 1 300.2.i.a 4
45.l even 12 1 inner 1620.2.x.b 8
72.j odd 6 1 960.2.v.e 4
72.l even 6 1 960.2.v.h 4
72.n even 6 1 960.2.v.e 4
72.p odd 6 1 960.2.v.h 4
180.n even 6 1 1200.2.v.i 4
180.p odd 6 1 1200.2.v.i 4
180.v odd 12 1 240.2.v.b 4
180.v odd 12 1 1200.2.v.i 4
180.x even 12 1 240.2.v.b 4
180.x even 12 1 1200.2.v.i 4
360.bo even 12 1 960.2.v.h 4
360.br even 12 1 960.2.v.e 4
360.bt odd 12 1 960.2.v.h 4
360.bu odd 12 1 960.2.v.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.i.a 4 9.c even 3 1
60.2.i.a 4 9.d odd 6 1
60.2.i.a 4 45.k odd 12 1
60.2.i.a 4 45.l even 12 1
240.2.v.b 4 36.f odd 6 1
240.2.v.b 4 36.h even 6 1
240.2.v.b 4 180.v odd 12 1
240.2.v.b 4 180.x even 12 1
300.2.i.a 4 45.h odd 6 1
300.2.i.a 4 45.j even 6 1
300.2.i.a 4 45.k odd 12 1
300.2.i.a 4 45.l even 12 1
960.2.v.e 4 72.j odd 6 1
960.2.v.e 4 72.n even 6 1
960.2.v.e 4 360.br even 12 1
960.2.v.e 4 360.bu odd 12 1
960.2.v.h 4 72.l even 6 1
960.2.v.h 4 72.p odd 6 1
960.2.v.h 4 360.bo even 12 1
960.2.v.h 4 360.bt odd 12 1
1200.2.v.i 4 180.n even 6 1
1200.2.v.i 4 180.p odd 6 1
1200.2.v.i 4 180.v odd 12 1
1200.2.v.i 4 180.x even 12 1
1620.2.x.b 8 1.a even 1 1 trivial
1620.2.x.b 8 3.b odd 2 1 inner
1620.2.x.b 8 5.c odd 4 1 inner
1620.2.x.b 8 9.c even 3 1 inner
1620.2.x.b 8 9.d odd 6 1 inner
1620.2.x.b 8 15.e even 4 1 inner
1620.2.x.b 8 45.k odd 12 1 inner
1620.2.x.b 8 45.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} - 4T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 100 T^{4} + 10000 \) Copy content Toggle raw display
$29$ \( (T^{4} + 20 T^{2} + 400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 18)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 80 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 8100 T^{4} + \cdots + 65610000 \) Copy content Toggle raw display
$53$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 80 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 36)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 8100 T^{4} + \cdots + 65610000 \) Copy content Toggle raw display
$89$ \( (T^{2} - 20)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 26244)^{2} \) Copy content Toggle raw display
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