Properties

Label 3600.3.l.m
Level $3600$
Weight $3$
Character orbit 3600.l
Analytic conductor $98.093$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3600,3,Mod(1601,3600)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3600.1601");
 
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.l (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: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 360)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 8) q^{7} + (3 \beta_{3} + \beta_1) q^{11} + (7 \beta_{2} - 2) q^{13} + (4 \beta_{3} - 8 \beta_1) q^{17} + (2 \beta_{2} - 24) q^{19} + ( - 5 \beta_{3} - 2 \beta_1) q^{23} + ( - \beta_{3} - 24 \beta_1) q^{29} + (6 \beta_{2} + 14) q^{31} + (11 \beta_{2} + 10) q^{37} + 15 \beta_1 q^{41} + (14 \beta_{2} - 28) q^{43} + ( - 6 \beta_{3} + 2 \beta_1) q^{47} + ( - 16 \beta_{2} + 25) q^{49} - 30 \beta_1 q^{53} + (19 \beta_{3} + \beta_1) q^{59} + ( - 14 \beta_{2} + 6) q^{61} + (8 \beta_{2} + 12) q^{67} + ( - 2 \beta_{3} + 20 \beta_1) q^{71} + (6 \beta_{2} + 90) q^{73} + ( - 23 \beta_{3} + 22 \beta_1) q^{77} + ( - 22 \beta_{2} - 50) q^{79} + (8 \beta_{3} + 62 \beta_1) q^{83} + (22 \beta_{3} - 25 \beta_1) q^{89} + ( - 58 \beta_{2} + 86) q^{91} + ( - 10 \beta_{2} - 78) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{7} - 8 q^{13} - 96 q^{19} + 56 q^{31} + 40 q^{37} - 112 q^{43} + 100 q^{49} + 24 q^{61} + 48 q^{67} + 360 q^{73} - 200 q^{79} + 344 q^{91} - 312 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 7\beta_1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
−1.58114 + 0.707107i
−1.58114 0.707107i
1.58114 0.707107i
1.58114 + 0.707107i
0 0 0 0 0 −11.1623 0 0 0
1601.2 0 0 0 0 0 −11.1623 0 0 0
1601.3 0 0 0 0 0 −4.83772 0 0 0
1601.4 0 0 0 0 0 −4.83772 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.3.l.m 4
3.b odd 2 1 inner 3600.3.l.m 4
4.b odd 2 1 1800.3.l.g 4
5.b even 2 1 720.3.l.d 4
5.c odd 4 2 3600.3.c.l 8
12.b even 2 1 1800.3.l.g 4
15.d odd 2 1 720.3.l.d 4
15.e even 4 2 3600.3.c.l 8
20.d odd 2 1 360.3.l.a 4
20.e even 4 2 1800.3.c.b 8
40.e odd 2 1 2880.3.l.a 4
40.f even 2 1 2880.3.l.h 4
60.h even 2 1 360.3.l.a 4
60.l odd 4 2 1800.3.c.b 8
120.i odd 2 1 2880.3.l.h 4
120.m even 2 1 2880.3.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.3.l.a 4 20.d odd 2 1
360.3.l.a 4 60.h even 2 1
720.3.l.d 4 5.b even 2 1
720.3.l.d 4 15.d odd 2 1
1800.3.c.b 8 20.e even 4 2
1800.3.c.b 8 60.l odd 4 2
1800.3.l.g 4 4.b odd 2 1
1800.3.l.g 4 12.b even 2 1
2880.3.l.a 4 40.e odd 2 1
2880.3.l.a 4 120.m even 2 1
2880.3.l.h 4 40.f even 2 1
2880.3.l.h 4 120.i odd 2 1
3600.3.c.l 8 5.c odd 4 2
3600.3.c.l 8 15.e even 4 2
3600.3.l.m 4 1.a even 1 1 trivial
3600.3.l.m 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{2} + 16T_{7} + 54 \) Copy content Toggle raw display
\( T_{11}^{4} + 364T_{11}^{2} + 31684 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 486 \) 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
$7$ \( (T^{2} + 16 T + 54)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 364 T^{2} + 31684 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 486)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 896 T^{2} + 36864 \) Copy content Toggle raw display
$19$ \( (T^{2} + 48 T + 536)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1016 T^{2} + 242064 \) Copy content Toggle raw display
$29$ \( T^{4} + 2344 T^{2} + \cdots + 1281424 \) Copy content Toggle raw display
$31$ \( (T^{2} - 28 T - 164)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 20 T - 1110)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 450)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 56 T - 1176)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1456 T^{2} + 506944 \) Copy content Toggle raw display
$53$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 14444 T^{2} + \cdots + 52099524 \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T - 1924)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 24 T - 496)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 1760 T^{2} + 518400 \) Copy content Toggle raw display
$73$ \( (T^{2} - 180 T + 7740)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 100 T - 2340)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 17936 T^{2} + \cdots + 41062464 \) Copy content Toggle raw display
$89$ \( T^{4} + 21860 T^{2} + \cdots + 71064900 \) Copy content Toggle raw display
$97$ \( (T^{2} + 156 T + 5084)^{2} \) Copy content Toggle raw display
show more
show less