Properties

Label 1200.2.v.e
Level $1200$
Weight $2$
Character orbit 1200.v
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,2,Mod(257,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.v (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + 2 \beta_1 q^{7} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + 2 \beta_1 q^{7} - 3 \beta_{2} q^{9} - 4 \beta_{3} q^{13} - 8 \beta_{2} q^{19} - 6 q^{21} + 3 \beta_1 q^{27} - 4 q^{31} + 4 \beta_1 q^{37} + 12 \beta_{2} q^{39} - 6 \beta_{3} q^{43} + 5 \beta_{2} q^{49} + 8 \beta_1 q^{57} + 14 q^{61} - 6 \beta_{3} q^{63} + 2 \beta_1 q^{67} + 8 \beta_{3} q^{73} - 4 \beta_{2} q^{79} - 9 q^{81} + 24 q^{91} - 4 \beta_{3} q^{93} + 8 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{21} - 16 q^{31} + 56 q^{61} - 36 q^{81} + 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 0 0 2.44949 + 2.44949i 0 3.00000i 0
257.2 0 1.22474 1.22474i 0 0 0 −2.44949 2.44949i 0 3.00000i 0
593.1 0 −1.22474 1.22474i 0 0 0 2.44949 2.44949i 0 3.00000i 0
593.2 0 1.22474 + 1.22474i 0 0 0 −2.44949 + 2.44949i 0 3.00000i 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 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.v.e 4
3.b odd 2 1 CM 1200.2.v.e 4
4.b odd 2 1 300.2.i.b 4
5.b even 2 1 inner 1200.2.v.e 4
5.c odd 4 2 inner 1200.2.v.e 4
12.b even 2 1 300.2.i.b 4
15.d odd 2 1 inner 1200.2.v.e 4
15.e even 4 2 inner 1200.2.v.e 4
20.d odd 2 1 300.2.i.b 4
20.e even 4 2 300.2.i.b 4
60.h even 2 1 300.2.i.b 4
60.l odd 4 2 300.2.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.i.b 4 4.b odd 2 1
300.2.i.b 4 12.b even 2 1
300.2.i.b 4 20.d odd 2 1
300.2.i.b 4 20.e even 4 2
300.2.i.b 4 60.h even 2 1
300.2.i.b 4 60.l odd 4 2
1200.2.v.e 4 1.a even 1 1 trivial
1200.2.v.e 4 3.b odd 2 1 CM
1200.2.v.e 4 5.b even 2 1 inner
1200.2.v.e 4 5.c odd 4 2 inner
1200.2.v.e 4 15.d odd 2 1 inner
1200.2.v.e 4 15.e even 4 2 inner

Hecke kernels

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

\( T_{7}^{4} + 144 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 144 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 2304 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 2304 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 11664 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T - 14)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 144 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 36864 \) Copy content Toggle raw display
$79$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 36864 \) Copy content Toggle raw display
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