Properties

Label 1200.2.o.b
Level $1200$
Weight $2$
Character orbit 1200.o
Analytic conductor $9.582$
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] = [1200,2,Mod(1199,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, 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("1200.1199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 240)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{3} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{3} q^{7} - 3 q^{9} + \beta_{3} q^{11} + 2 \beta_1 q^{13} + 6 q^{17} + 2 \beta_{2} q^{19} - 3 \beta_1 q^{21} + 2 \beta_{2} q^{23} + 3 \beta_{2} q^{27} - 3 \beta_1 q^{29} + 2 \beta_{2} q^{31} - 3 \beta_1 q^{33} + 2 \beta_1 q^{37} + 2 \beta_{3} q^{39} - 6 \beta_1 q^{41} + 2 \beta_{3} q^{43} + 2 \beta_{2} q^{47} + 5 q^{49} - 6 \beta_{2} q^{51} - 6 q^{53} + 6 q^{57} - \beta_{3} q^{59} - 10 q^{61} - 3 \beta_{3} q^{63} - 2 \beta_{3} q^{67} + 6 q^{69} + 4 \beta_{3} q^{71} - \beta_1 q^{73} + 12 q^{77} - 6 \beta_{2} q^{79} + 9 q^{81} - 6 \beta_{2} q^{83} - 3 \beta_{3} q^{87} + 8 \beta_{2} q^{91} + 6 q^{93} - 5 \beta_1 q^{97} - 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 24 q^{17} + 20 q^{49} - 24 q^{53} + 24 q^{57} - 40 q^{61} + 24 q^{69} + 48 q^{77} + 36 q^{81} + 24 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1199.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 1.73205i 0 0 0 −3.46410 0 −3.00000 0
1199.2 0 1.73205i 0 0 0 3.46410 0 −3.00000 0
1199.3 0 1.73205i 0 0 0 −3.46410 0 −3.00000 0
1199.4 0 1.73205i 0 0 0 3.46410 0 −3.00000 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
15.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 1200.2.o.b 4
3.b odd 2 1 1200.2.o.a 4
4.b odd 2 1 inner 1200.2.o.b 4
5.b even 2 1 1200.2.o.a 4
5.c odd 4 1 240.2.h.b 4
5.c odd 4 1 1200.2.h.m 4
12.b even 2 1 1200.2.o.a 4
15.d odd 2 1 inner 1200.2.o.b 4
15.e even 4 1 240.2.h.b 4
15.e even 4 1 1200.2.h.m 4
20.d odd 2 1 1200.2.o.a 4
20.e even 4 1 240.2.h.b 4
20.e even 4 1 1200.2.h.m 4
40.i odd 4 1 960.2.h.d 4
40.k even 4 1 960.2.h.d 4
60.h even 2 1 inner 1200.2.o.b 4
60.l odd 4 1 240.2.h.b 4
60.l odd 4 1 1200.2.h.m 4
120.q odd 4 1 960.2.h.d 4
120.w even 4 1 960.2.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.h.b 4 5.c odd 4 1
240.2.h.b 4 15.e even 4 1
240.2.h.b 4 20.e even 4 1
240.2.h.b 4 60.l odd 4 1
960.2.h.d 4 40.i odd 4 1
960.2.h.d 4 40.k even 4 1
960.2.h.d 4 120.q odd 4 1
960.2.h.d 4 120.w even 4 1
1200.2.h.m 4 5.c odd 4 1
1200.2.h.m 4 15.e even 4 1
1200.2.h.m 4 20.e even 4 1
1200.2.h.m 4 60.l odd 4 1
1200.2.o.a 4 3.b odd 2 1
1200.2.o.a 4 5.b even 2 1
1200.2.o.a 4 12.b even 2 1
1200.2.o.a 4 20.d odd 2 1
1200.2.o.b 4 1.a even 1 1 trivial
1200.2.o.b 4 4.b odd 2 1 inner
1200.2.o.b 4 15.d odd 2 1 inner
1200.2.o.b 4 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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