Properties

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

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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: \(\Q(\sqrt{-2}, \sqrt{19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 20x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1800)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{7} + \beta_1 q^{11} + (2 \beta_{3} - 5) q^{13} + ( - \beta_{2} + \beta_1) q^{17} + (\beta_{3} - 3) q^{19} + (2 \beta_{2} - 5 \beta_1) q^{23} + (\beta_{2} - 3 \beta_1) q^{29} + ( - 3 \beta_{3} - 19) q^{31} + ( - 2 \beta_{3} + 28) q^{37} + ( - 3 \beta_{2} - 18 \beta_1) q^{41} + (\beta_{3} + 23) q^{43} + ( - 3 \beta_{2} + 29 \beta_1) q^{47} + ( - 2 \beta_{3} + 28) q^{49} + (3 \beta_{2} + 18 \beta_1) q^{53} + ( - \beta_{2} + 25 \beta_1) q^{59} + ( - 4 \beta_{3} + 57) q^{61} + (\beta_{3} - 27) q^{67} + ( - 7 \beta_{2} + 38 \beta_1) q^{71} + ( - 6 \beta_{3} - 36) q^{73} + ( - \beta_{2} + \beta_1) q^{77} + ( - 8 \beta_{3} - 8) q^{79} + ( - 5 \beta_{2} - 13 \beta_1) q^{83} + ( - 10 \beta_{2} + 20 \beta_1) q^{89} + (7 \beta_{3} - 157) q^{91} + (4 \beta_{3} + 111) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} - 20 q^{13} - 12 q^{19} - 76 q^{31} + 112 q^{37} + 92 q^{43} + 112 q^{49} + 228 q^{61} - 108 q^{67} - 144 q^{73} - 32 q^{79} - 628 q^{91} + 444 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 20x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 11\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 58\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{2} + 58\beta_1 ) / 4 \) 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
2.37510i
2.37510i
3.78931i
3.78931i
0 0 0 0 0 −7.71780 0 0 0
1601.2 0 0 0 0 0 −7.71780 0 0 0
1601.3 0 0 0 0 0 9.71780 0 0 0
1601.4 0 0 0 0 0 9.71780 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.t 4
3.b odd 2 1 inner 3600.3.l.t 4
4.b odd 2 1 1800.3.l.c 4
5.b even 2 1 3600.3.l.p 4
5.c odd 4 2 3600.3.c.j 8
12.b even 2 1 1800.3.l.c 4
15.d odd 2 1 3600.3.l.p 4
15.e even 4 2 3600.3.c.j 8
20.d odd 2 1 1800.3.l.e yes 4
20.e even 4 2 1800.3.c.c 8
60.h even 2 1 1800.3.l.e yes 4
60.l odd 4 2 1800.3.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.3.c.c 8 20.e even 4 2
1800.3.c.c 8 60.l odd 4 2
1800.3.l.c 4 4.b odd 2 1
1800.3.l.c 4 12.b even 2 1
1800.3.l.e yes 4 20.d odd 2 1
1800.3.l.e yes 4 60.h even 2 1
3600.3.c.j 8 5.c odd 4 2
3600.3.c.j 8 15.e even 4 2
3600.3.l.p 4 5.b even 2 1
3600.3.l.p 4 15.d odd 2 1
3600.3.l.t 4 1.a even 1 1 trivial
3600.3.l.t 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 \(S_{3}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{2} - 2T_{7} - 75 \) Copy content Toggle raw display
\( T_{11}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 10T_{13} - 279 \) 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
$7$ \( (T^{2} - 2 T - 75)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 10 T - 279)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 308 T^{2} + 22500 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 67)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1316 T^{2} + 311364 \) Copy content Toggle raw display
$29$ \( T^{4} + 340 T^{2} + 17956 \) Copy content Toggle raw display
$31$ \( (T^{2} + 38 T - 323)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 56 T + 480)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 4032 T^{2} + 518400 \) Copy content Toggle raw display
$43$ \( (T^{2} - 46 T + 453)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6100 T^{2} + 98596 \) Copy content Toggle raw display
$53$ \( T^{4} + 4032 T^{2} + 518400 \) Copy content Toggle raw display
$59$ \( T^{4} + 2804 T^{2} + \cdots + 1205604 \) Copy content Toggle raw display
$61$ \( (T^{2} - 114 T + 2033)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 54 T + 653)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 20672 T^{2} + \cdots + 20793600 \) Copy content Toggle raw display
$73$ \( (T^{2} + 72 T - 1440)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 16 T - 4800)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 8276 T^{2} + \cdots + 11985444 \) Copy content Toggle raw display
$89$ \( T^{4} + 32000 T^{2} + \cdots + 207360000 \) Copy content Toggle raw display
$97$ \( (T^{2} - 222 T + 11105)^{2} \) Copy content Toggle raw display
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