Properties

Label 2700.3.b.i
Level $2700$
Weight $3$
Character orbit 2700.b
Analytic conductor $73.570$
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] = [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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q - 4 \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_1 q^{7} - \beta_{2} q^{11} + 7 \beta_1 q^{13} + 3 \beta_{3} q^{17} - 8 q^{19} - \beta_{3} q^{23} + 7 \beta_{2} q^{29} + 29 q^{31} + 2 \beta_1 q^{37} - 2 \beta_{2} q^{41} + 7 \beta_1 q^{43} + 5 \beta_{3} q^{47} + 33 q^{49} - 14 \beta_{3} q^{53} - 6 \beta_{2} q^{59} + 62 q^{61} - 58 \beta_1 q^{67} - 8 \beta_{2} q^{71} + 52 \beta_1 q^{73} - 4 \beta_{3} q^{77} + 49 q^{79} + 2 \beta_{3} q^{83} + 16 \beta_{2} q^{89} + 28 q^{91} - 34 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{19} + 116 q^{31} + 132 q^{49} + 248 q^{61} + 196 q^{79} + 112 q^{91}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 6\nu^{2} + 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 9 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 6\beta_1 ) / 3 \) 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
0.618034i
1.61803i
1.61803i
0.618034i
0 0 0 0 0 4.00000i 0 0 0
1349.2 0 0 0 0 0 4.00000i 0 0 0
1349.3 0 0 0 0 0 4.00000i 0 0 0
1349.4 0 0 0 0 0 4.00000i 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
5.b even 2 1 inner
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.i 4
3.b odd 2 1 inner 2700.3.b.i 4
5.b even 2 1 inner 2700.3.b.i 4
5.c odd 4 1 540.3.g.b 2
5.c odd 4 1 2700.3.g.k 2
15.d odd 2 1 inner 2700.3.b.i 4
15.e even 4 1 540.3.g.b 2
15.e even 4 1 2700.3.g.k 2
20.e even 4 1 2160.3.l.c 2
45.k odd 12 2 1620.3.o.c 4
45.l even 12 2 1620.3.o.c 4
60.l odd 4 1 2160.3.l.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.b 2 5.c odd 4 1
540.3.g.b 2 15.e even 4 1
1620.3.o.c 4 45.k odd 12 2
1620.3.o.c 4 45.l even 12 2
2160.3.l.c 2 20.e even 4 1
2160.3.l.c 2 60.l odd 4 1
2700.3.b.i 4 1.a even 1 1 trivial
2700.3.b.i 4 3.b odd 2 1 inner
2700.3.b.i 4 5.b even 2 1 inner
2700.3.b.i 4 15.d odd 2 1 inner
2700.3.g.k 2 5.c odd 4 1
2700.3.g.k 2 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 \(S_{3}^{\mathrm{new}}(2700, [\chi])\):

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 45 \) Copy content Toggle raw display
\( T_{13}^{2} + 49 \) Copy content Toggle raw display
\( T_{17}^{2} - 405 \) 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
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 405)^{2} \) Copy content Toggle raw display
$19$ \( (T + 8)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2205)^{2} \) Copy content Toggle raw display
$31$ \( (T - 29)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1125)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8820)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1620)^{2} \) Copy content Toggle raw display
$61$ \( (T - 62)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 3364)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2880)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2704)^{2} \) Copy content Toggle raw display
$79$ \( (T - 49)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 11520)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
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