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Properties

Label 1620.3.o.c
Level $1620$
Weight $3$
Character orbit 1620.o
Analytic conductor $44.142$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,3,Mod(701,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.701");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.o (of order \(6\), degree \(2\), 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: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + 4 \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + 4 \beta_{2} q^{7} + (3 \beta_{3} - 3 \beta_1) q^{11} + ( - 7 \beta_{2} + 7) q^{13} - 9 \beta_{3} q^{17} + 8 q^{19} + 3 \beta_1 q^{23} + 5 \beta_{2} q^{25} + (21 \beta_{3} - 21 \beta_1) q^{29} + (29 \beta_{2} - 29) q^{31} - 4 \beta_{3} q^{35} + 2 q^{37} + 6 \beta_1 q^{41} + 7 \beta_{2} q^{43} + (15 \beta_{3} - 15 \beta_1) q^{47} + ( - 33 \beta_{2} + 33) q^{49} - 42 \beta_{3} q^{53} + 15 q^{55} - 18 \beta_1 q^{59} - 62 \beta_{2} q^{61} + (7 \beta_{3} - 7 \beta_1) q^{65} + ( - 58 \beta_{2} + 58) q^{67} - 24 \beta_{3} q^{71} - 52 q^{73} - 12 \beta_1 q^{77} + 49 \beta_{2} q^{79} + ( - 6 \beta_{3} + 6 \beta_1) q^{83} + (45 \beta_{2} - 45) q^{85} - 48 \beta_{3} q^{89} + 28 q^{91} - 8 \beta_1 q^{95} + 34 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} + 14 q^{13} + 32 q^{19} + 10 q^{25} - 58 q^{31} + 8 q^{37} + 14 q^{43} + 66 q^{49} + 60 q^{55} - 124 q^{61} + 116 q^{67} - 208 q^{73} + 98 q^{79} - 90 q^{85} + 112 q^{91} + 68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
1.93649 + 1.11803i
−1.93649 − 1.11803i
1.93649 − 1.11803i
−1.93649 + 1.11803i
0 0 0 −1.93649 − 1.11803i 0 2.00000 + 3.46410i 0 0 0
701.2 0 0 0 1.93649 + 1.11803i 0 2.00000 + 3.46410i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 2.00000 − 3.46410i 0 0 0
1241.2 0 0 0 1.93649 − 1.11803i 0 2.00000 − 3.46410i 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
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.o.c 4
3.b odd 2 1 inner 1620.3.o.c 4
9.c even 3 1 540.3.g.b ✓ 2
9.c even 3 1 inner 1620.3.o.c 4
9.d odd 6 1 540.3.g.b ✓ 2
9.d odd 6 1 inner 1620.3.o.c 4
36.f odd 6 1 2160.3.l.c 2
36.h even 6 1 2160.3.l.c 2
45.h odd 6 1 2700.3.g.k 2
45.j even 6 1 2700.3.g.k 2
45.k odd 12 2 2700.3.b.i 4
45.l even 12 2 2700.3.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.b ✓ 2 9.c even 3 1
540.3.g.b ✓ 2 9.d odd 6 1
1620.3.o.c 4 1.a even 1 1 trivial
1620.3.o.c 4 3.b odd 2 1 inner
1620.3.o.c 4 9.c even 3 1 inner
1620.3.o.c 4 9.d odd 6 1 inner
2160.3.l.c 2 36.f odd 6 1
2160.3.l.c 2 36.h even 6 1
2700.3.b.i 4 45.k odd 12 2
2700.3.b.i 4 45.l even 12 2
2700.3.g.k 2 45.h odd 6 1
2700.3.g.k 2 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 4T_{7} + 16 \) acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 45T^{2} + 2025 \) Copy content Toggle raw display
$13$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 405)^{2} \) Copy content Toggle raw display
$19$ \( (T - 8)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 45T^{2} + 2025 \) Copy content Toggle raw display
$29$ \( T^{4} - 2205 T^{2} + 4862025 \) Copy content Toggle raw display
$31$ \( (T^{2} + 29 T + 841)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 180 T^{2} + 32400 \) Copy content Toggle raw display
$43$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 1125 T^{2} + 1265625 \) Copy content Toggle raw display
$53$ \( (T^{2} + 8820)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 1620 T^{2} + 2624400 \) Copy content Toggle raw display
$61$ \( (T^{2} + 62 T + 3844)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 58 T + 3364)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2880)^{2} \) Copy content Toggle raw display
$73$ \( (T + 52)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 49 T + 2401)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 180 T^{2} + 32400 \) Copy content Toggle raw display
$89$ \( (T^{2} + 11520)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 34 T + 1156)^{2} \) Copy content Toggle raw display
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