Properties

Label 900.2.k.i
Level $900$
Weight $2$
Character orbit 900.k
Analytic conductor $7.187$
Analytic rank $0$
Dimension $8$
CM discriminant -15
Inner twists $16$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(307,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.k (of order \(4\), degree \(2\), 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{4} + \beta_{2}) q^{4} + ( - \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_{4} + \beta_{2}) q^{4} + ( - \beta_{3} + \beta_1) q^{8} + ( - \beta_{6} + 3) q^{16} + (2 \beta_{7} + 4 \beta_{5}) q^{17} + ( - 2 \beta_{4} + 4 \beta_{2}) q^{19} + ( - 4 \beta_{3} - 2 \beta_1) q^{23} + ( - 4 \beta_{6} - 2) q^{31} + ( - \beta_{7} + 3 \beta_{5}) q^{32} + (4 \beta_{4} + 4 \beta_{2}) q^{34} + ( - 2 \beta_{3} + 4 \beta_1) q^{38} + ( - 4 \beta_{6} - 12) q^{46} + (2 \beta_{7} - 4 \beta_{5}) q^{47} + 7 \beta_{4} q^{49} + (8 \beta_{3} - 4 \beta_1) q^{53} - 2 q^{61} + ( - 4 \beta_{7} - 2 \beta_{5}) q^{62} + ( - 7 \beta_{4} + 3 \beta_{2}) q^{64} + (4 \beta_{3} + 4 \beta_1) q^{68} + ( - 2 \beta_{6} + 14) q^{76} + ( - 2 \beta_{4} + 4 \beta_{2}) q^{79} + ( - 8 \beta_{3} - 4 \beta_1) q^{83} + ( - 4 \beta_{7} - 12 \beta_{5}) q^{92} + (12 \beta_{4} - 4 \beta_{2}) q^{94} + 7 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 28 q^{16} - 80 q^{46} - 16 q^{61} + 120 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 3\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 7\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{4} - 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 3\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 4\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} + 12\beta_{5} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−1.40294 0.178197i
−0.178197 1.40294i
0.178197 + 1.40294i
1.40294 + 0.178197i
−1.40294 + 0.178197i
−0.178197 + 1.40294i
0.178197 1.40294i
1.40294 0.178197i
−1.40294 + 0.178197i 0 1.93649 0.500000i 0 0 0 −2.62769 + 1.04655i 0 0
307.2 −0.178197 + 1.40294i 0 −1.93649 0.500000i 0 0 0 1.04655 2.62769i 0 0
307.3 0.178197 1.40294i 0 −1.93649 0.500000i 0 0 0 −1.04655 + 2.62769i 0 0
307.4 1.40294 0.178197i 0 1.93649 0.500000i 0 0 0 2.62769 1.04655i 0 0
343.1 −1.40294 0.178197i 0 1.93649 + 0.500000i 0 0 0 −2.62769 1.04655i 0 0
343.2 −0.178197 1.40294i 0 −1.93649 + 0.500000i 0 0 0 1.04655 + 2.62769i 0 0
343.3 0.178197 + 1.40294i 0 −1.93649 + 0.500000i 0 0 0 −1.04655 2.62769i 0 0
343.4 1.40294 + 0.178197i 0 1.93649 + 0.500000i 0 0 0 2.62769 + 1.04655i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
12.b even 2 1 inner
15.e even 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner
60.h even 2 1 inner
60.l odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.k.i 8
3.b odd 2 1 inner 900.2.k.i 8
4.b odd 2 1 inner 900.2.k.i 8
5.b even 2 1 inner 900.2.k.i 8
5.c odd 4 2 inner 900.2.k.i 8
12.b even 2 1 inner 900.2.k.i 8
15.d odd 2 1 CM 900.2.k.i 8
15.e even 4 2 inner 900.2.k.i 8
20.d odd 2 1 inner 900.2.k.i 8
20.e even 4 2 inner 900.2.k.i 8
60.h even 2 1 inner 900.2.k.i 8
60.l odd 4 2 inner 900.2.k.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.k.i 8 1.a even 1 1 trivial
900.2.k.i 8 3.b odd 2 1 inner
900.2.k.i 8 4.b odd 2 1 inner
900.2.k.i 8 5.b even 2 1 inner
900.2.k.i 8 5.c odd 4 2 inner
900.2.k.i 8 12.b even 2 1 inner
900.2.k.i 8 15.d odd 2 1 CM
900.2.k.i 8 15.e even 4 2 inner
900.2.k.i 8 20.d odd 2 1 inner
900.2.k.i 8 20.e even 4 2 inner
900.2.k.i 8 60.h even 2 1 inner
900.2.k.i 8 60.l odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{4} + 2304 \) Copy content Toggle raw display
\( T_{19}^{2} - 60 \) Copy content Toggle raw display
\( T_{23}^{4} + 6400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 6400)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 36864)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T + 2)^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 102400)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
show more
show less