Properties

Label 900.3.b.a
Level $900$
Weight $3$
Character orbit 900.b
Analytic conductor $24.523$
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] = [900,3,Mod(449,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, 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("900.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{7} - \beta_{2} q^{11} - 7 \beta_1 q^{13} + \beta_{3} q^{17} + 7 q^{19} - 7 \beta_{3} q^{23} - 7 \beta_{2} q^{29} + 17 q^{31} + 16 \beta_1 q^{37} - 12 \beta_{2} q^{41} - 55 \beta_1 q^{43} + 11 \beta_{3} q^{47} + 48 q^{49} - 20 \beta_{3} q^{53} - 13 \beta_{2} q^{59} + 65 q^{61} + 49 \beta_1 q^{67} - 12 \beta_{2} q^{71} - 88 \beta_1 q^{73} + \beta_{3} q^{77} + 40 q^{79} - 37 \beta_{3} q^{83} - 24 \beta_{2} q^{89} + 7 q^{91} - 41 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{19} + 68 q^{31} + 192 q^{49} + 260 q^{61} + 160 q^{79} + 28 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{8}^{3} + 3\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\zeta_{8}^{3} + 3\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 0 0 1.00000i 0 0 0
449.2 0 0 0 0 0 1.00000i 0 0 0
449.3 0 0 0 0 0 1.00000i 0 0 0
449.4 0 0 0 0 0 1.00000i 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 900.3.b.a 4
3.b odd 2 1 inner 900.3.b.a 4
4.b odd 2 1 3600.3.c.d 4
5.b even 2 1 inner 900.3.b.a 4
5.c odd 4 1 900.3.g.a 2
5.c odd 4 1 900.3.g.b yes 2
12.b even 2 1 3600.3.c.d 4
15.d odd 2 1 inner 900.3.b.a 4
15.e even 4 1 900.3.g.a 2
15.e even 4 1 900.3.g.b yes 2
20.d odd 2 1 3600.3.c.d 4
20.e even 4 1 3600.3.l.f 2
20.e even 4 1 3600.3.l.g 2
60.h even 2 1 3600.3.c.d 4
60.l odd 4 1 3600.3.l.f 2
60.l odd 4 1 3600.3.l.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.b.a 4 1.a even 1 1 trivial
900.3.b.a 4 3.b odd 2 1 inner
900.3.b.a 4 5.b even 2 1 inner
900.3.b.a 4 15.d odd 2 1 inner
900.3.g.a 2 5.c odd 4 1
900.3.g.a 2 15.e even 4 1
900.3.g.b yes 2 5.c odd 4 1
900.3.g.b yes 2 15.e even 4 1
3600.3.c.d 4 4.b odd 2 1
3600.3.c.d 4 12.b even 2 1
3600.3.c.d 4 20.d odd 2 1
3600.3.c.d 4 60.h even 2 1
3600.3.l.f 2 20.e even 4 1
3600.3.l.f 2 60.l odd 4 1
3600.3.l.g 2 20.e even 4 1
3600.3.l.g 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\). 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$19$ \( (T - 7)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 882)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 882)^{2} \) Copy content Toggle raw display
$31$ \( (T - 17)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3025)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2178)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 7200)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 3042)^{2} \) Copy content Toggle raw display
$61$ \( (T - 65)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2401)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 7744)^{2} \) Copy content Toggle raw display
$79$ \( (T - 40)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 24642)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 10368)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1681)^{2} \) Copy content Toggle raw display
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