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Properties

Label 1200.3.bg.b
Level $1200$
Weight $3$
Character orbit 1200.bg
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(193,1200)]
 
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, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.bg (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: \(32.6976317232\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (2 \beta_{3} + 4 \beta_{2} - 4) q^{7} + 3 \beta_{2} q^{9} + ( - 4 \beta_{3} + 4 \beta_1 + 2) q^{11} - 2 \beta_1 q^{13} + (4 \beta_{3} - 4 \beta_{2} + 4) q^{17} + (8 \beta_{3} + 14 \beta_{2} + 8 \beta_1) q^{19}+ \cdots + (12 \beta_{3} + 6 \beta_{2} + 12 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7} + 8 q^{11} + 16 q^{17} - 24 q^{21} - 48 q^{23} - 56 q^{31} + 48 q^{33} + 32 q^{37} - 8 q^{41} + 128 q^{43} + 80 q^{47} - 48 q^{51} - 32 q^{53} - 96 q^{57} - 120 q^{61} - 48 q^{63} - 96 q^{67}+ \cdots + 160 q^{97}+O(q^{100}) \) 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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) 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

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−1.22474 − 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 − 1.22474i
0 −1.22474 − 1.22474i 0 0 0 −1.55051 + 1.55051i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −6.44949 + 6.44949i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −1.55051 − 1.55051i 0 − 3.00000i 0
1057.2 0 1.22474 − 1.22474i 0 0 0 −6.44949 − 6.44949i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.bg.b 4
4.b odd 2 1 600.3.u.f yes 4
5.b even 2 1 1200.3.bg.m 4
5.c odd 4 1 inner 1200.3.bg.b 4
5.c odd 4 1 1200.3.bg.m 4
12.b even 2 1 1800.3.v.q 4
20.d odd 2 1 600.3.u.a ✓ 4
20.e even 4 1 600.3.u.a ✓ 4
20.e even 4 1 600.3.u.f yes 4
60.h even 2 1 1800.3.v.j 4
60.l odd 4 1 1800.3.v.j 4
60.l odd 4 1 1800.3.v.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.u.a ✓ 4 20.d odd 2 1
600.3.u.a ✓ 4 20.e even 4 1
600.3.u.f yes 4 4.b odd 2 1
600.3.u.f yes 4 20.e even 4 1
1200.3.bg.b 4 1.a even 1 1 trivial
1200.3.bg.b 4 5.c odd 4 1 inner
1200.3.bg.m 4 5.b even 2 1
1200.3.bg.m 4 5.c odd 4 1
1800.3.v.j 4 60.h even 2 1
1800.3.v.j 4 60.l odd 4 1
1800.3.v.q 4 12.b even 2 1
1800.3.v.q 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 16T_{7}^{3} + 128T_{7}^{2} + 320T_{7} + 400 \) acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\). 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
$7$ \( T^{4} + 16 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 92)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 144 \) Copy content Toggle raw display
$17$ \( T^{4} - 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{4} + 1160 T^{2} + 35344 \) Copy content Toggle raw display
$23$ \( T^{4} + 48 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$29$ \( T^{4} + 1280 T^{2} + 16384 \) Copy content Toggle raw display
$31$ \( (T^{2} + 28 T - 1340)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 32 T^{3} + \cdots + 211600 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 3452)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 128 T^{3} + \cdots + 3444736 \) Copy content Toggle raw display
$47$ \( T^{4} - 80 T^{3} + \cdots + 160000 \) Copy content Toggle raw display
$53$ \( T^{4} + 32 T^{3} + \cdots + 14137600 \) Copy content Toggle raw display
$59$ \( T^{4} + 7688 T^{2} + 913936 \) Copy content Toggle raw display
$61$ \( (T^{2} + 60 T - 5244)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 96 T^{3} + \cdots + 21678336 \) Copy content Toggle raw display
$71$ \( (T^{2} - 16 T - 3392)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 256 T^{3} + \cdots + 60217600 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4356)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 160 T^{3} + \cdots + 473344 \) Copy content Toggle raw display
$89$ \( T^{4} + 968 T^{2} + 80656 \) Copy content Toggle raw display
$97$ \( T^{4} - 160 T^{3} + \cdots + 199600384 \) Copy content Toggle raw display
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