Properties

Label 7200.2.h.g
Level $7200$
Weight $2$
Character orbit 7200.h
Analytic conductor $57.492$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7200,2,Mod(1151,7200)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.1151");
 
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.h (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: \(57.4922894553\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{7} + (\beta_{3} + 4) q^{11} + ( - 4 \beta_{3} - 1) q^{13} + (\beta_{2} + 2 \beta_1) q^{17} + ( - 3 \beta_{2} - 3 \beta_1) q^{19} + ( - \beta_{3} + 4) q^{23} + ( - 3 \beta_{2} - 6 \beta_1) q^{29} + (3 \beta_{2} + \beta_1) q^{31} + ( - 2 \beta_{3} - 4) q^{37} + ( - 2 \beta_{2} - 2 \beta_1) q^{41} + (\beta_{2} - 5 \beta_1) q^{43} + ( - \beta_{3} - 6) q^{47} + ( - 2 \beta_{3} + 4) q^{49} + ( - 2 \beta_{2} + 2 \beta_1) q^{53} + ( - 5 \beta_{3} + 2) q^{59} + ( - 2 \beta_{3} + 1) q^{61} + ( - 3 \beta_{2} + \beta_1) q^{67} + ( - 4 \beta_{3} - 6) q^{71} + (6 \beta_{3} + 8) q^{73} + (5 \beta_{2} + 6 \beta_1) q^{77} - 12 \beta_1 q^{79} + (\beta_{3} - 6) q^{83} + 12 \beta_1 q^{89} + ( - 5 \beta_{2} - 9 \beta_1) q^{91} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{11} - 4 q^{13} + 16 q^{23} - 16 q^{37} - 24 q^{47} + 16 q^{49} + 8 q^{59} + 4 q^{61} - 24 q^{71} + 32 q^{73} - 24 q^{83} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) 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
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 0 0 0 0 2.41421i 0 0 0
1151.2 0 0 0 0 0 0.414214i 0 0 0
1151.3 0 0 0 0 0 0.414214i 0 0 0
1151.4 0 0 0 0 0 2.41421i 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
12.b 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.h.g yes 4
3.b odd 2 1 7200.2.h.a 4
4.b odd 2 1 7200.2.h.a 4
5.b even 2 1 7200.2.h.h yes 4
5.c odd 4 1 7200.2.o.g 4
5.c odd 4 1 7200.2.o.i 4
12.b even 2 1 inner 7200.2.h.g yes 4
15.d odd 2 1 7200.2.h.b yes 4
15.e even 4 1 7200.2.o.f 4
15.e even 4 1 7200.2.o.h 4
20.d odd 2 1 7200.2.h.b yes 4
20.e even 4 1 7200.2.o.f 4
20.e even 4 1 7200.2.o.h 4
60.h even 2 1 7200.2.h.h yes 4
60.l odd 4 1 7200.2.o.g 4
60.l odd 4 1 7200.2.o.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7200.2.h.a 4 3.b odd 2 1
7200.2.h.a 4 4.b odd 2 1
7200.2.h.b yes 4 15.d odd 2 1
7200.2.h.b yes 4 20.d odd 2 1
7200.2.h.g yes 4 1.a even 1 1 trivial
7200.2.h.g yes 4 12.b even 2 1 inner
7200.2.h.h yes 4 5.b even 2 1
7200.2.h.h yes 4 60.h even 2 1
7200.2.o.f 4 15.e even 4 1
7200.2.o.f 4 20.e even 4 1
7200.2.o.g 4 5.c odd 4 1
7200.2.o.g 4 60.l odd 4 1
7200.2.o.h 4 15.e even 4 1
7200.2.o.h 4 20.e even 4 1
7200.2.o.i 4 5.c odd 4 1
7200.2.o.i 4 60.l odd 4 1

Hecke kernels

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

\( T_{7}^{4} + 6T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 8T_{11} + 14 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 31 \) Copy content Toggle raw display
\( T_{23}^{2} - 8T_{23} + 14 \) 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
$7$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8 T + 14)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 31)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 54T^{2} + 81 \) Copy content Toggle raw display
$23$ \( (T^{2} - 8 T + 14)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 108T^{2} + 324 \) Copy content Toggle raw display
$31$ \( T^{4} + 38T^{2} + 289 \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 54T^{2} + 529 \) Copy content Toggle raw display
$47$ \( (T^{2} + 12 T + 34)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 46)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 38T^{2} + 289 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 16 T - 8)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 34)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$97$ \( (T + 9)^{4} \) Copy content Toggle raw display
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