Properties

Label 7200.2.o.a
Level $7200$
Weight $2$
Character orbit 7200.o
Analytic conductor $57.492$
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] = [7200,2,Mod(7199,7200)]
 
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([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.7199");
 
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.o (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: \(4\)
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: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 288)
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.

\(f(q)\) \(=\) \( q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{7} + 4 \beta_{3} q^{11} + 2 \beta_1 q^{13} + 3 \beta_{3} q^{17} - 4 \beta_{2} q^{23} + \beta_{2} q^{29} - 2 \beta_1 q^{31} + 3 \beta_1 q^{37} - 7 \beta_{2} q^{41} - 8 q^{43} + 4 \beta_{2} q^{47} + 9 q^{49} - 3 \beta_{3} q^{53} + 8 \beta_{3} q^{59} - 2 q^{61} + 8 q^{67} + 4 \beta_{3} q^{71} - 16 \beta_{3} q^{77} - 2 \beta_1 q^{79} - 4 \beta_{2} q^{83} + 3 \beta_{2} q^{89} - 8 \beta_1 q^{91} + 4 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{7} - 32 q^{43} + 36 q^{49} - 8 q^{61} + 32 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) 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
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) 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 \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7199.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 0 0 −4.00000 0 0 0
7199.2 0 0 0 0 0 −4.00000 0 0 0
7199.3 0 0 0 0 0 −4.00000 0 0 0
7199.4 0 0 0 0 0 −4.00000 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
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.o.a 4
3.b odd 2 1 inner 7200.2.o.a 4
4.b odd 2 1 7200.2.o.n 4
5.b even 2 1 7200.2.o.n 4
5.c odd 4 1 288.2.c.a 4
5.c odd 4 1 7200.2.h.d 4
12.b even 2 1 7200.2.o.n 4
15.d odd 2 1 7200.2.o.n 4
15.e even 4 1 288.2.c.a 4
15.e even 4 1 7200.2.h.d 4
20.d odd 2 1 inner 7200.2.o.a 4
20.e even 4 1 288.2.c.a 4
20.e even 4 1 7200.2.h.d 4
40.i odd 4 1 576.2.c.c 4
40.k even 4 1 576.2.c.c 4
45.k odd 12 2 2592.2.s.d 8
45.l even 12 2 2592.2.s.d 8
60.h even 2 1 inner 7200.2.o.a 4
60.l odd 4 1 288.2.c.a 4
60.l odd 4 1 7200.2.h.d 4
80.i odd 4 1 2304.2.f.c 4
80.j even 4 1 2304.2.f.c 4
80.s even 4 1 2304.2.f.e 4
80.t odd 4 1 2304.2.f.e 4
120.q odd 4 1 576.2.c.c 4
120.w even 4 1 576.2.c.c 4
180.v odd 12 2 2592.2.s.d 8
180.x even 12 2 2592.2.s.d 8
240.z odd 4 1 2304.2.f.e 4
240.bb even 4 1 2304.2.f.c 4
240.bd odd 4 1 2304.2.f.c 4
240.bf even 4 1 2304.2.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.c.a 4 5.c odd 4 1
288.2.c.a 4 15.e even 4 1
288.2.c.a 4 20.e even 4 1
288.2.c.a 4 60.l odd 4 1
576.2.c.c 4 40.i odd 4 1
576.2.c.c 4 40.k even 4 1
576.2.c.c 4 120.q odd 4 1
576.2.c.c 4 120.w even 4 1
2304.2.f.c 4 80.i odd 4 1
2304.2.f.c 4 80.j even 4 1
2304.2.f.c 4 240.bb even 4 1
2304.2.f.c 4 240.bd odd 4 1
2304.2.f.e 4 80.s even 4 1
2304.2.f.e 4 80.t odd 4 1
2304.2.f.e 4 240.z odd 4 1
2304.2.f.e 4 240.bf even 4 1
2592.2.s.d 8 45.k odd 12 2
2592.2.s.d 8 45.l even 12 2
2592.2.s.d 8 180.v odd 12 2
2592.2.s.d 8 180.x even 12 2
7200.2.h.d 4 5.c odd 4 1
7200.2.h.d 4 15.e even 4 1
7200.2.h.d 4 20.e even 4 1
7200.2.h.d 4 60.l odd 4 1
7200.2.o.a 4 1.a even 1 1 trivial
7200.2.o.a 4 3.b odd 2 1 inner
7200.2.o.a 4 20.d odd 2 1 inner
7200.2.o.a 4 60.h even 2 1 inner
7200.2.o.n 4 4.b odd 2 1
7200.2.o.n 4 5.b even 2 1
7200.2.o.n 4 12.b even 2 1
7200.2.o.n 4 15.d odd 2 1

Hecke kernels

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

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 32 \) Copy content Toggle raw display
\( T_{17}^{2} - 18 \) 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
$7$ \( (T + 4)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{4} \) Copy content Toggle raw display
$67$ \( (T - 8)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
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