Properties

Label 3600.3.e.t
Level $3600$
Weight $3$
Character orbit 3600.e
Analytic conductor $98.093$
Analytic rank $0$
Dimension $2$
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(3151,3600)]
 
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([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3600.3151");
 
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.e (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 48)
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 \(\beta = 4\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{7} + 3 \beta q^{11} + 14 q^{13} - 6 q^{17} - \beta q^{19} - 30 q^{29} + 3 \beta q^{31} - 26 q^{37} + 54 q^{41} + 3 \beta q^{43} + 6 \beta q^{47} + q^{49} - 18 q^{53} - 3 \beta q^{59} - 70 q^{61} - 17 \beta q^{67} + 12 \beta q^{71} - 82 q^{73} + 144 q^{77} + 11 \beta q^{79} + 3 \beta q^{83} - 114 q^{89} - 14 \beta q^{91} - 34 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 28 q^{13} - 12 q^{17} - 60 q^{29} - 52 q^{37} + 108 q^{41} + 2 q^{49} - 36 q^{53} - 140 q^{61} - 164 q^{73} + 288 q^{77} - 228 q^{89} - 68 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3151.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 6.92820i 0 0 0
3151.2 0 0 0 0 0 6.92820i 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
4.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.e.t 2
3.b odd 2 1 1200.3.e.h 2
4.b odd 2 1 inner 3600.3.e.t 2
5.b even 2 1 144.3.g.b 2
5.c odd 4 2 3600.3.j.i 4
12.b even 2 1 1200.3.e.h 2
15.d odd 2 1 48.3.g.a 2
15.e even 4 2 1200.3.j.a 4
20.d odd 2 1 144.3.g.b 2
20.e even 4 2 3600.3.j.i 4
40.e odd 2 1 576.3.g.i 2
40.f even 2 1 576.3.g.i 2
45.h odd 6 1 1296.3.o.a 2
45.h odd 6 1 1296.3.o.c 2
45.j even 6 1 1296.3.o.n 2
45.j even 6 1 1296.3.o.p 2
60.h even 2 1 48.3.g.a 2
60.l odd 4 2 1200.3.j.a 4
80.k odd 4 2 2304.3.b.n 4
80.q even 4 2 2304.3.b.n 4
105.g even 2 1 2352.3.m.a 2
120.i odd 2 1 192.3.g.a 2
120.m even 2 1 192.3.g.a 2
180.n even 6 1 1296.3.o.a 2
180.n even 6 1 1296.3.o.c 2
180.p odd 6 1 1296.3.o.n 2
180.p odd 6 1 1296.3.o.p 2
240.t even 4 2 768.3.b.b 4
240.bm odd 4 2 768.3.b.b 4
420.o odd 2 1 2352.3.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.g.a 2 15.d odd 2 1
48.3.g.a 2 60.h even 2 1
144.3.g.b 2 5.b even 2 1
144.3.g.b 2 20.d odd 2 1
192.3.g.a 2 120.i odd 2 1
192.3.g.a 2 120.m even 2 1
576.3.g.i 2 40.e odd 2 1
576.3.g.i 2 40.f even 2 1
768.3.b.b 4 240.t even 4 2
768.3.b.b 4 240.bm odd 4 2
1200.3.e.h 2 3.b odd 2 1
1200.3.e.h 2 12.b even 2 1
1200.3.j.a 4 15.e even 4 2
1200.3.j.a 4 60.l odd 4 2
1296.3.o.a 2 45.h odd 6 1
1296.3.o.a 2 180.n even 6 1
1296.3.o.c 2 45.h odd 6 1
1296.3.o.c 2 180.n even 6 1
1296.3.o.n 2 45.j even 6 1
1296.3.o.n 2 180.p odd 6 1
1296.3.o.p 2 45.j even 6 1
1296.3.o.p 2 180.p odd 6 1
2304.3.b.n 4 80.k odd 4 2
2304.3.b.n 4 80.q even 4 2
2352.3.m.a 2 105.g even 2 1
2352.3.m.a 2 420.o odd 2 1
3600.3.e.t 2 1.a even 1 1 trivial
3600.3.e.t 2 4.b odd 2 1 inner
3600.3.j.i 4 5.c odd 4 2
3600.3.j.i 4 20.e even 4 2

Hecke kernels

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

\( T_{7}^{2} + 48 \) Copy content Toggle raw display
\( T_{11}^{2} + 432 \) Copy content Toggle raw display
\( T_{13} - 14 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 48 \) Copy content Toggle raw display
$11$ \( T^{2} + 432 \) Copy content Toggle raw display
$13$ \( (T - 14)^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 48 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 30)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 432 \) Copy content Toggle raw display
$37$ \( (T + 26)^{2} \) Copy content Toggle raw display
$41$ \( (T - 54)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 432 \) Copy content Toggle raw display
$47$ \( T^{2} + 1728 \) Copy content Toggle raw display
$53$ \( (T + 18)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 432 \) Copy content Toggle raw display
$61$ \( (T + 70)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 13872 \) Copy content Toggle raw display
$71$ \( T^{2} + 6912 \) Copy content Toggle raw display
$73$ \( (T + 82)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 5808 \) Copy content Toggle raw display
$83$ \( T^{2} + 432 \) Copy content Toggle raw display
$89$ \( (T + 114)^{2} \) Copy content Toggle raw display
$97$ \( (T + 34)^{2} \) Copy content Toggle raw display
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