↑

Properties

Label 1200.3.bg.d
Level $1200$
Weight $3$
Character orbit 1200.bg
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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 30)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{7} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{7} - 3 \beta_{2} q^{9} + ( - 4 \beta_{3} + 4 \beta_1 + 4) q^{11} + (8 \beta_{3} + 3 \beta_{2} - 3) q^{13} + ( - 11 \beta_{2} - 4 \beta_1 - 11) q^{17} + ( - 4 \beta_{3} - 16 \beta_{2} - 4 \beta_1) q^{19} + (4 \beta_{3} - 4 \beta_1 + 12) q^{21} + ( - 12 \beta_{3} + 4 \beta_{2} - 4) q^{23} - 3 \beta_1 q^{27} + ( - 4 \beta_{3} - 16 \beta_{2} - 4 \beta_1) q^{29} + ( - 8 \beta_{3} + 8 \beta_1 + 20) q^{31} + ( - 4 \beta_{3} - 12 \beta_{2} + 12) q^{33} + (27 \beta_{2} + 27) q^{37} + (3 \beta_{3} + 24 \beta_{2} + 3 \beta_1) q^{39} + ( - 4 \beta_{3} + 4 \beta_1 + 8) q^{41} + (20 \beta_{3} - 12 \beta_{2} + 12) q^{43} + (24 \beta_{2} - 12 \beta_1 + 24) q^{47} + ( - 32 \beta_{3} + 31 \beta_{2} - 32 \beta_1) q^{49} + (11 \beta_{3} - 11 \beta_1 - 12) q^{51} + ( - 36 \beta_{3} + 25 \beta_{2} - 25) q^{53} + ( - 12 \beta_{2} - 16 \beta_1 - 12) q^{57} - 20 \beta_{2} q^{59} + ( - 16 \beta_{3} + 16 \beta_1 - 24) q^{61} + ( - 12 \beta_{3} + 12 \beta_{2} - 12) q^{63} + ( - 4 \beta_{2} - 36 \beta_1 - 4) q^{67} + (4 \beta_{3} - 36 \beta_{2} + 4 \beta_1) q^{69} + ( - 4 \beta_{3} + 4 \beta_1 - 16) q^{71} + (8 \beta_{3} - 47 \beta_{2} + 47) q^{73} + (32 \beta_{2} - 16 \beta_1 + 32) q^{77} + ( - 52 \beta_{3} + 12 \beta_{2} - 52 \beta_1) q^{79} - 9 q^{81} + ( - 28 \beta_{3} - 48 \beta_{2} + 48) q^{83} + ( - 12 \beta_{2} - 16 \beta_1 - 12) q^{87} + (12 \beta_{3} + 88 \beta_{2} + 12 \beta_1) q^{89} + ( - 20 \beta_{3} + 20 \beta_1 - 72) q^{91} + ( - 20 \beta_{3} - 24 \beta_{2} + 24) q^{93} + ( - 33 \beta_{2} - 40 \beta_1 - 33) q^{97} + ( - 12 \beta_{3} - 12 \beta_{2} - 12 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{7} + 16 q^{11} - 12 q^{13} - 44 q^{17} + 48 q^{21} - 16 q^{23} + 80 q^{31} + 48 q^{33} + 108 q^{37} + 32 q^{41} + 48 q^{43} + 96 q^{47} - 48 q^{51} - 100 q^{53} - 48 q^{57} - 96 q^{61} - 48 q^{63} - 16 q^{67} - 64 q^{71} + 188 q^{73} + 128 q^{77} - 36 q^{81} + 192 q^{83} - 48 q^{87} - 288 q^{91} + 96 q^{93} - 132 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −8.89898 + 8.89898i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 0.898979 − 0.898979i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −8.89898 − 8.89898i 0 − 3.00000i 0
1057.2 0 1.22474 − 1.22474i 0 0 0 0.898979 + 0.898979i 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.d 4
4.b odd 2 1 150.3.f.b 4
5.b even 2 1 240.3.bg.b 4
5.c odd 4 1 240.3.bg.b 4
5.c odd 4 1 inner 1200.3.bg.d 4
12.b even 2 1 450.3.g.j 4
15.d odd 2 1 720.3.bh.i 4
15.e even 4 1 720.3.bh.i 4
20.d odd 2 1 30.3.f.a ✓ 4
20.e even 4 1 30.3.f.a ✓ 4
20.e even 4 1 150.3.f.b 4
40.e odd 2 1 960.3.bg.e 4
40.f even 2 1 960.3.bg.g 4
40.i odd 4 1 960.3.bg.g 4
40.k even 4 1 960.3.bg.e 4
60.h even 2 1 90.3.g.d 4
60.l odd 4 1 90.3.g.d 4
60.l odd 4 1 450.3.g.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.f.a ✓ 4 20.d odd 2 1
30.3.f.a ✓ 4 20.e even 4 1
90.3.g.d 4 60.h even 2 1
90.3.g.d 4 60.l odd 4 1
150.3.f.b 4 4.b odd 2 1
150.3.f.b 4 20.e even 4 1
240.3.bg.b 4 5.b even 2 1
240.3.bg.b 4 5.c odd 4 1
450.3.g.j 4 12.b even 2 1
450.3.g.j 4 60.l odd 4 1
720.3.bh.i 4 15.d odd 2 1
720.3.bh.i 4 15.e even 4 1
960.3.bg.e 4 40.e odd 2 1
960.3.bg.e 4 40.k even 4 1
960.3.bg.g 4 40.f even 2 1
960.3.bg.g 4 40.i odd 4 1
1200.3.bg.d 4 1.a even 1 1 trivial
1200.3.bg.d 4 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 16T_{7}^{3} + 128T_{7}^{2} - 256T_{7} + 256 \) 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
$7$ \( T^{4} + 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8 T - 80)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} + \cdots + 30276 \) Copy content Toggle raw display
$17$ \( T^{4} + 44 T^{3} + \cdots + 37636 \) Copy content Toggle raw display
$19$ \( T^{4} + 704 T^{2} + 25600 \) Copy content Toggle raw display
$23$ \( T^{4} + 16 T^{3} + \cdots + 160000 \) Copy content Toggle raw display
$29$ \( T^{4} + 704 T^{2} + 25600 \) Copy content Toggle raw display
$31$ \( (T^{2} - 40 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 54 T + 1458)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 48 T^{3} + \cdots + 831744 \) Copy content Toggle raw display
$47$ \( T^{4} - 96 T^{3} + \cdots + 518400 \) Copy content Toggle raw display
$53$ \( T^{4} + 100 T^{3} + \cdots + 6959044 \) Copy content Toggle raw display
$59$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48 T - 960)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + \cdots + 14868736 \) Copy content Toggle raw display
$71$ \( (T^{2} + 32 T + 160)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 188 T^{3} + \cdots + 17859076 \) Copy content Toggle raw display
$79$ \( T^{4} + 32736 T^{2} + 258566400 \) Copy content Toggle raw display
$83$ \( T^{4} - 192 T^{3} + \cdots + 5089536 \) Copy content Toggle raw display
$89$ \( T^{4} + 17216 T^{2} + 47334400 \) Copy content Toggle raw display
$97$ \( T^{4} + 132 T^{3} + \cdots + 6874884 \) Copy content Toggle raw display
show more
show less