Properties

Label 2700.3.b.g
Level $2700$
Weight $3$
Character orbit 2700.b
Analytic conductor $73.570$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2700,3,Mod(1349,2700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2700.1349");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2700.b (of order \(2\), degree \(1\), 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: \(73.5696713773\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 540)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 2 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 2 \beta_1) q^{7} + (\beta_{2} - \beta_1) q^{11} + ( - \beta_{2} - 4 \beta_1) q^{13} - 15 q^{17} + (\beta_{3} + 1) q^{19} + (\beta_{3} - 21) q^{23} + ( - \beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{3} - 7) q^{31} + (6 \beta_{2} + 11 \beta_1) q^{37} + (5 \beta_{2} + 7 \beta_1) q^{41} + ( - 7 \beta_{2} + 11 \beta_1) q^{43} + (\beta_{3} - 12) q^{47} + (5 \beta_{3} - 21) q^{49} + (5 \beta_{3} + 3) q^{53} + ( - 3 \beta_{2} + 24 \beta_1) q^{59} + ( - 6 \beta_{3} - 1) q^{61} - 7 \beta_1 q^{67} + ( - 7 \beta_{2} + 19 \beta_1) q^{71} + ( - 3 \beta_{2} - 5 \beta_1) q^{73} + (4 \beta_{3} - 60) q^{77} + ( - 6 \beta_{3} + 67) q^{79} + ( - 8 \beta_{3} + 33) q^{83} + (17 \beta_{2} + 13 \beta_1) q^{89} + (\beta_{3} + 10) q^{91} + 2 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 60 q^{17} + 4 q^{19} - 84 q^{23} - 28 q^{31} - 48 q^{47} - 84 q^{49} + 12 q^{53} - 4 q^{61} - 240 q^{77} + 268 q^{79} + 132 q^{83} + 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 12\nu^{2} + 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 18 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{2} + 5\beta_1 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1349.1
1.61803i
0.618034i
0.618034i
1.61803i
0 0 0 0 0 11.7082i 0 0 0
1349.2 0 0 0 0 0 1.70820i 0 0 0
1349.3 0 0 0 0 0 1.70820i 0 0 0
1349.4 0 0 0 0 0 11.7082i 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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.3.b.g 4
3.b odd 2 1 2700.3.b.l 4
5.b even 2 1 2700.3.b.l 4
5.c odd 4 1 540.3.g.d 4
5.c odd 4 1 2700.3.g.n 4
15.d odd 2 1 inner 2700.3.b.g 4
15.e even 4 1 540.3.g.d 4
15.e even 4 1 2700.3.g.n 4
20.e even 4 1 2160.3.l.e 4
45.k odd 12 2 1620.3.o.e 8
45.l even 12 2 1620.3.o.e 8
60.l odd 4 1 2160.3.l.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.d 4 5.c odd 4 1
540.3.g.d 4 15.e even 4 1
1620.3.o.e 8 45.k odd 12 2
1620.3.o.e 8 45.l even 12 2
2160.3.l.e 4 20.e even 4 1
2160.3.l.e 4 60.l odd 4 1
2700.3.b.g 4 1.a even 1 1 trivial
2700.3.b.g 4 15.d odd 2 1 inner
2700.3.b.l 4 3.b odd 2 1
2700.3.b.l 4 5.b even 2 1
2700.3.g.n 4 5.c odd 4 1
2700.3.g.n 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2700, [\chi])\):

\( T_{7}^{4} + 140T_{7}^{2} + 400 \) Copy content Toggle raw display
\( T_{11}^{4} + 108T_{11}^{2} + 1296 \) Copy content Toggle raw display
\( T_{13}^{4} + 188T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17} + 15 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 140T^{2} + 400 \) Copy content Toggle raw display
$11$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( T^{4} + 188T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T + 15)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 179)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 42 T + 261)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$31$ \( (T^{2} + 14 T - 131)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 3752 T^{2} + \cdots + 1860496 \) Copy content Toggle raw display
$41$ \( T^{4} + 2412 T^{2} + \cdots + 1089936 \) Copy content Toggle raw display
$43$ \( T^{4} + 6092 T^{2} + \cdots + 1860496 \) Copy content Toggle raw display
$47$ \( (T^{2} + 24 T - 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T - 4491)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 6012 T^{2} + \cdots + 4822416 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 6479)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 8460 T^{2} + 32400 \) Copy content Toggle raw display
$73$ \( T^{4} + 908 T^{2} + 126736 \) Copy content Toggle raw display
$79$ \( (T^{2} - 134 T - 1991)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 66 T - 10431)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 26172 T^{2} + \cdots + 167029776 \) Copy content Toggle raw display
$97$ \( T^{4} + 368 T^{2} + 30976 \) Copy content Toggle raw display
show more
show less