Properties

Label 3600.3.c.e
Level $3600$
Weight $3$
Character orbit 3600.c
Analytic conductor $98.093$
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] = [3600,3,Mod(449,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3600.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3600.c (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: \(98.0928951697\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 225)
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.

\(f(q)\) \(=\) \( q - 9 \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 \beta_1 q^{7} - 13 \beta_{2} q^{11} + \beta_1 q^{13} + 19 \beta_{3} q^{17} + 25 q^{19} - 3 \beta_{3} q^{23} + 19 \beta_{2} q^{29} + 39 q^{31} - 32 \beta_1 q^{37} - 4 \beta_{2} q^{41} + 23 \beta_1 q^{43} + 23 \beta_{3} q^{47} - 32 q^{49} + 68 \beta_{3} q^{53} + 7 \beta_{2} q^{59} + 73 q^{61} + 63 \beta_1 q^{67} - 44 \beta_{2} q^{71} + 136 \beta_1 q^{73} - 117 \beta_{3} q^{77} - 24 q^{79} - 33 \beta_{3} q^{83} - 72 \beta_{2} q^{89} + 9 q^{91} + 7 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 100 q^{19} + 156 q^{31} - 128 q^{49} + 292 q^{61} - 96 q^{79} + 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) 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 \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 9.00000i 0 0 0
449.2 0 0 0 0 0 9.00000i 0 0 0
449.3 0 0 0 0 0 9.00000i 0 0 0
449.4 0 0 0 0 0 9.00000i 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 3600.3.c.e 4
3.b odd 2 1 inner 3600.3.c.e 4
4.b odd 2 1 225.3.d.a 4
5.b even 2 1 inner 3600.3.c.e 4
5.c odd 4 1 3600.3.l.b 2
5.c odd 4 1 3600.3.l.j 2
12.b even 2 1 225.3.d.a 4
15.d odd 2 1 inner 3600.3.c.e 4
15.e even 4 1 3600.3.l.b 2
15.e even 4 1 3600.3.l.j 2
20.d odd 2 1 225.3.d.a 4
20.e even 4 1 225.3.c.a 2
20.e even 4 1 225.3.c.b yes 2
60.h even 2 1 225.3.d.a 4
60.l odd 4 1 225.3.c.a 2
60.l odd 4 1 225.3.c.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.3.c.a 2 20.e even 4 1
225.3.c.a 2 60.l odd 4 1
225.3.c.b yes 2 20.e even 4 1
225.3.c.b yes 2 60.l odd 4 1
225.3.d.a 4 4.b odd 2 1
225.3.d.a 4 12.b even 2 1
225.3.d.a 4 20.d odd 2 1
225.3.d.a 4 60.h even 2 1
3600.3.c.e 4 1.a even 1 1 trivial
3600.3.c.e 4 3.b odd 2 1 inner
3600.3.c.e 4 5.b even 2 1 inner
3600.3.c.e 4 15.d odd 2 1 inner
3600.3.l.b 2 5.c odd 4 1
3600.3.l.b 2 15.e even 4 1
3600.3.l.j 2 5.c odd 4 1
3600.3.l.j 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 \(S_{3}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{2} + 81 \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) 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
$7$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 338)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 722)^{2} \) Copy content Toggle raw display
$19$ \( (T - 25)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 722)^{2} \) Copy content Toggle raw display
$31$ \( (T - 39)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 529)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1058)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 9248)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$61$ \( (T - 73)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 3969)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 3872)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 18496)^{2} \) Copy content Toggle raw display
$79$ \( (T + 24)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2178)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 10368)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
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