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Properties

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

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2} - 2) q^{7} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_{2} - 2) q^{7} + 3 \beta_{2} q^{9} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{11} + (12 \beta_{2} + 5 \beta_1 + 12) q^{13} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{17} + (2 \beta_{3} + 5 \beta_{2} + 2 \beta_1) q^{19} + (2 \beta_{3} - 2 \beta_1 - 3) q^{21} + ( - 18 \beta_{2} + 6 \beta_1 - 18) q^{23} + 3 \beta_{3} q^{27} + ( - 14 \beta_{3} - 2 \beta_{2} - 14 \beta_1) q^{29} + (6 \beta_{3} - 6 \beta_1 - 11) q^{31} + (6 \beta_{2} + 2 \beta_1 + 6) q^{33} + (12 \beta_{3} - 28 \beta_{2} + 28) q^{37} + (12 \beta_{3} + 15 \beta_{2} + 12 \beta_1) q^{39} + ( - 2 \beta_{3} + 2 \beta_1 - 8) q^{41} + ( - 26 \beta_{2} + 19 \beta_1 - 26) q^{43} + (14 \beta_{3} + 20 \beta_{2} - 20) q^{47} + ( - 4 \beta_{3} + 38 \beta_{2} - 4 \beta_1) q^{49} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{51} + (26 \beta_{2} + 32 \beta_1 + 26) q^{53} + (5 \beta_{3} + 6 \beta_{2} - 6) q^{57} + (32 \beta_{3} + 2 \beta_{2} + 32 \beta_1) q^{59} + ( - 24 \beta_{3} + 24 \beta_1 - 45) q^{61} + ( - 6 \beta_{2} - 3 \beta_1 - 6) q^{63} + (15 \beta_{3} + 66 \beta_{2} - 66) q^{67} + ( - 18 \beta_{3} + 18 \beta_{2} - 18 \beta_1) q^{69} + (14 \beta_{3} - 14 \beta_1 - 64) q^{71} + (28 \beta_{2} + 20 \beta_1 + 28) q^{73} + (10 \beta_{3} + 10 \beta_{2} - 10) q^{77} + ( - 4 \beta_{3} + 78 \beta_{2} - 4 \beta_1) q^{79} - 9 q^{81} + (4 \beta_{2} + 70 \beta_1 + 4) q^{83} + ( - 2 \beta_{3} - 42 \beta_{2} + 42) q^{87} + ( - 28 \beta_{3} + 28 \beta_{2} - 28 \beta_1) q^{89} + (22 \beta_{3} - 22 \beta_1 - 63) q^{91} + ( - 18 \beta_{2} - 11 \beta_1 - 18) q^{93} + ( - 21 \beta_{3} - 80 \beta_{2} + 80) q^{97} + (6 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} + 8 q^{11} + 48 q^{13} + 8 q^{17} - 12 q^{21} - 72 q^{23} - 44 q^{31} + 24 q^{33} + 112 q^{37} - 32 q^{41} - 104 q^{43} - 80 q^{47} - 24 q^{51} + 104 q^{53} - 24 q^{57} - 180 q^{61} - 24 q^{63} - 264 q^{67} - 256 q^{71} + 112 q^{73} - 40 q^{77} - 36 q^{81} + 16 q^{83} + 168 q^{87} - 252 q^{91} - 72 q^{93} + 320 q^{97}+O(q^{100}) \) 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 −0.775255 + 0.775255i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −3.22474 + 3.22474i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −0.775255 − 0.775255i 0 − 3.00000i 0
1057.2 0 1.22474 − 1.22474i 0 0 0 −3.22474 − 3.22474i 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.f 4
4.b odd 2 1 600.3.u.d yes 4
5.b even 2 1 1200.3.bg.l 4
5.c odd 4 1 inner 1200.3.bg.f 4
5.c odd 4 1 1200.3.bg.l 4
12.b even 2 1 1800.3.v.m 4
20.d odd 2 1 600.3.u.c ✓ 4
20.e even 4 1 600.3.u.c ✓ 4
20.e even 4 1 600.3.u.d yes 4
60.h even 2 1 1800.3.v.l 4
60.l odd 4 1 1800.3.v.l 4
60.l odd 4 1 1800.3.v.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.u.c ✓ 4 20.d odd 2 1
600.3.u.c ✓ 4 20.e even 4 1
600.3.u.d yes 4 4.b odd 2 1
600.3.u.d yes 4 20.e even 4 1
1200.3.bg.f 4 1.a even 1 1 trivial
1200.3.bg.f 4 5.c odd 4 1 inner
1200.3.bg.l 4 5.b even 2 1
1200.3.bg.l 4 5.c odd 4 1
1800.3.v.l 4 60.h even 2 1
1800.3.v.l 4 60.l odd 4 1
1800.3.v.m 4 12.b even 2 1
1800.3.v.m 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 8T_{7}^{3} + 32T_{7}^{2} + 40T_{7} + 25 \) 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
$7$ \( T^{4} + 8 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 48 T^{3} + \cdots + 45369 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 98T^{2} + 1 \) Copy content Toggle raw display
$23$ \( T^{4} + 72 T^{3} + \cdots + 291600 \) Copy content Toggle raw display
$29$ \( T^{4} + 2360 T^{2} + 1373584 \) Copy content Toggle raw display
$31$ \( (T^{2} + 22 T - 95)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 112 T^{3} + \cdots + 1290496 \) Copy content Toggle raw display
$41$ \( (T^{2} + 16 T + 40)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 104 T^{3} + \cdots + 72361 \) Copy content Toggle raw display
$47$ \( T^{4} + 80 T^{3} + \cdots + 44944 \) Copy content Toggle raw display
$53$ \( T^{4} - 104 T^{3} + \cdots + 2958400 \) Copy content Toggle raw display
$59$ \( T^{4} + 12296 T^{2} + 37699600 \) Copy content Toggle raw display
$61$ \( (T^{2} + 90 T - 1431)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 264 T^{3} + \cdots + 64593369 \) Copy content Toggle raw display
$71$ \( (T^{2} + 128 T + 2920)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 112 T^{3} + \cdots + 135424 \) Copy content Toggle raw display
$79$ \( T^{4} + 12360 T^{2} + 35856144 \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + \cdots + 215150224 \) Copy content Toggle raw display
$89$ \( T^{4} + 10976 T^{2} + 15366400 \) Copy content Toggle raw display
$97$ \( T^{4} - 320 T^{3} + \cdots + 131721529 \) Copy content Toggle raw display
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