Properties

Label 900.2.j.a
Level $900$
Weight $2$
Character orbit 900.j
Analytic conductor $7.187$
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] = [900,2,Mod(557,900)]
 
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([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.557");
 
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.j (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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_1 - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_1 - 2) q^{7} + 2 \beta_{2} q^{11} + ( - 3 \beta_1 - 3) q^{13} + (\beta_{3} + \beta_{2}) q^{17} - 4 \beta_1 q^{19} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{23} - 7 \beta_{3} q^{29} - 8 q^{31} + ( - 3 \beta_1 + 3) q^{37} - \beta_{2} q^{41} + (6 \beta_{3} + 6 \beta_{2}) q^{47} - \beta_1 q^{49} + ( - 5 \beta_{3} + 5 \beta_{2}) q^{53} + 2 \beta_{3} q^{59} + (8 \beta_1 - 8) q^{67} + 4 \beta_{2} q^{71} + ( - 7 \beta_1 - 7) q^{73} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{77} + (6 \beta_{3} - 6 \beta_{2}) q^{83} + \beta_{3} q^{89} + 12 q^{91} + (3 \beta_1 - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 12 q^{13} - 32 q^{31} + 12 q^{37} - 32 q^{67} - 28 q^{73} + 48 q^{91} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 0 0 −2.00000 2.00000i 0 0 0
557.2 0 0 0 0 0 −2.00000 2.00000i 0 0 0
593.1 0 0 0 0 0 −2.00000 + 2.00000i 0 0 0
593.2 0 0 0 0 0 −2.00000 + 2.00000i 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.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.j.a 4
3.b odd 2 1 inner 900.2.j.a 4
4.b odd 2 1 3600.2.w.f 4
5.b even 2 1 180.2.j.a 4
5.c odd 4 1 180.2.j.a 4
5.c odd 4 1 inner 900.2.j.a 4
12.b even 2 1 3600.2.w.f 4
15.d odd 2 1 180.2.j.a 4
15.e even 4 1 180.2.j.a 4
15.e even 4 1 inner 900.2.j.a 4
20.d odd 2 1 720.2.w.b 4
20.e even 4 1 720.2.w.b 4
20.e even 4 1 3600.2.w.f 4
40.e odd 2 1 2880.2.w.a 4
40.f even 2 1 2880.2.w.j 4
40.i odd 4 1 2880.2.w.j 4
40.k even 4 1 2880.2.w.a 4
45.h odd 6 2 1620.2.x.a 8
45.j even 6 2 1620.2.x.a 8
45.k odd 12 2 1620.2.x.a 8
45.l even 12 2 1620.2.x.a 8
60.h even 2 1 720.2.w.b 4
60.l odd 4 1 720.2.w.b 4
60.l odd 4 1 3600.2.w.f 4
120.i odd 2 1 2880.2.w.j 4
120.m even 2 1 2880.2.w.a 4
120.q odd 4 1 2880.2.w.a 4
120.w even 4 1 2880.2.w.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.j.a 4 5.b even 2 1
180.2.j.a 4 5.c odd 4 1
180.2.j.a 4 15.d odd 2 1
180.2.j.a 4 15.e even 4 1
720.2.w.b 4 20.d odd 2 1
720.2.w.b 4 20.e even 4 1
720.2.w.b 4 60.h even 2 1
720.2.w.b 4 60.l odd 4 1
900.2.j.a 4 1.a even 1 1 trivial
900.2.j.a 4 3.b odd 2 1 inner
900.2.j.a 4 5.c odd 4 1 inner
900.2.j.a 4 15.e even 4 1 inner
1620.2.x.a 8 45.h odd 6 2
1620.2.x.a 8 45.j even 6 2
1620.2.x.a 8 45.k odd 12 2
1620.2.x.a 8 45.l even 12 2
2880.2.w.a 4 40.e odd 2 1
2880.2.w.a 4 40.k even 4 1
2880.2.w.a 4 120.m even 2 1
2880.2.w.a 4 120.q odd 4 1
2880.2.w.j 4 40.f even 2 1
2880.2.w.j 4 40.i odd 4 1
2880.2.w.j 4 120.i odd 2 1
2880.2.w.j 4 120.w even 4 1
3600.2.w.f 4 4.b odd 2 1
3600.2.w.f 4 12.b even 2 1
3600.2.w.f 4 20.e even 4 1
3600.2.w.f 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4T_{7} + 8 \) acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\). 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
$7$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 4096 \) Copy content Toggle raw display
$29$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 20736 \) Copy content Toggle raw display
$53$ \( T^{4} + 10000 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16 T + 128)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 20736 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
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