Properties

Label 7200.2.f.n
Level $7200$
Weight $2$
Character orbit 7200.f
Analytic conductor $57.492$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7200,2,Mod(6049,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.6049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.f (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: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta q^{13} + 4 \beta q^{17} - 4 q^{29} + \beta q^{37} + 8 q^{41} + 7 q^{49} + 2 \beta q^{53} - 10 q^{61} + 3 \beta q^{73} + 16 q^{89} + 9 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{29} + 16 q^{41} + 14 q^{49} - 20 q^{61} + 32 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6049.1
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
6049.2 0 0 0 0 0 0 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 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.f.n 2
3.b odd 2 1 7200.2.f.q 2
4.b odd 2 1 CM 7200.2.f.n 2
5.b even 2 1 inner 7200.2.f.n 2
5.c odd 4 1 288.2.a.a 1
5.c odd 4 1 7200.2.a.bf 1
12.b even 2 1 7200.2.f.q 2
15.d odd 2 1 7200.2.f.q 2
15.e even 4 1 288.2.a.e yes 1
15.e even 4 1 7200.2.a.be 1
20.d odd 2 1 inner 7200.2.f.n 2
20.e even 4 1 288.2.a.a 1
20.e even 4 1 7200.2.a.bf 1
40.i odd 4 1 576.2.a.i 1
40.k even 4 1 576.2.a.i 1
45.k odd 12 2 2592.2.i.x 2
45.l even 12 2 2592.2.i.a 2
60.h even 2 1 7200.2.f.q 2
60.l odd 4 1 288.2.a.e yes 1
60.l odd 4 1 7200.2.a.be 1
80.i odd 4 1 2304.2.d.h 2
80.j even 4 1 2304.2.d.h 2
80.s even 4 1 2304.2.d.h 2
80.t odd 4 1 2304.2.d.h 2
120.q odd 4 1 576.2.a.a 1
120.w even 4 1 576.2.a.a 1
180.v odd 12 2 2592.2.i.a 2
180.x even 12 2 2592.2.i.x 2
240.z odd 4 1 2304.2.d.l 2
240.bb even 4 1 2304.2.d.l 2
240.bd odd 4 1 2304.2.d.l 2
240.bf even 4 1 2304.2.d.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.a.a 1 5.c odd 4 1
288.2.a.a 1 20.e even 4 1
288.2.a.e yes 1 15.e even 4 1
288.2.a.e yes 1 60.l odd 4 1
576.2.a.a 1 120.q odd 4 1
576.2.a.a 1 120.w even 4 1
576.2.a.i 1 40.i odd 4 1
576.2.a.i 1 40.k even 4 1
2304.2.d.h 2 80.i odd 4 1
2304.2.d.h 2 80.j even 4 1
2304.2.d.h 2 80.s even 4 1
2304.2.d.h 2 80.t odd 4 1
2304.2.d.l 2 240.z odd 4 1
2304.2.d.l 2 240.bb even 4 1
2304.2.d.l 2 240.bd odd 4 1
2304.2.d.l 2 240.bf even 4 1
2592.2.i.a 2 45.l even 12 2
2592.2.i.a 2 180.v odd 12 2
2592.2.i.x 2 45.k odd 12 2
2592.2.i.x 2 180.x even 12 2
7200.2.a.be 1 15.e even 4 1
7200.2.a.be 1 60.l odd 4 1
7200.2.a.bf 1 5.c odd 4 1
7200.2.a.bf 1 20.e even 4 1
7200.2.f.n 2 1.a even 1 1 trivial
7200.2.f.n 2 4.b odd 2 1 CM
7200.2.f.n 2 5.b even 2 1 inner
7200.2.f.n 2 20.d odd 2 1 inner
7200.2.f.q 2 3.b odd 2 1
7200.2.f.q 2 12.b even 2 1
7200.2.f.q 2 15.d odd 2 1
7200.2.f.q 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} + 64 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{29} + 4 \) Copy content Toggle raw display
\( T_{31} \) 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
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 64 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 324 \) Copy content Toggle raw display
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