Properties

Label 1200.4.f.x
Level $1200$
Weight $4$
Character orbit 1200.f
Analytic conductor $70.802$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(49,1200)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (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: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{109})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 55x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 600)
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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{7} - 9 q^{9} + (3 \beta_{3} + 8) q^{11} + ( - 2 \beta_{2} + 41 \beta_1) q^{13} + ( - 5 \beta_{2} - 4 \beta_1) q^{17} + ( - \beta_{3} - 105) q^{19} + (3 \beta_{3} - 3) q^{21} + ( - \beta_{2} - 20 \beta_1) q^{23} - 27 \beta_1 q^{27} + (\beta_{3} + 120) q^{29} + ( - 9 \beta_{3} - 109) q^{31} + (9 \beta_{2} + 24 \beta_1) q^{33} + (10 \beta_{2} - 182 \beta_1) q^{37} + (6 \beta_{3} - 123) q^{39} + ( - 3 \beta_{3} + 42) q^{41} + (5 \beta_{2} - 137 \beta_1) q^{43} + ( - 12 \beta_{2} + 262 \beta_1) q^{47} + (2 \beta_{3} - 94) q^{49} + (15 \beta_{3} + 12) q^{51} + ( - 3 \beta_{2} + 222 \beta_1) q^{53} + ( - 3 \beta_{2} - 315 \beta_1) q^{57} + (8 \beta_{3} - 542) q^{59} + ( - 4 \beta_{3} + 387) q^{61} + (9 \beta_{2} - 9 \beta_1) q^{63} + (37 \beta_{2} + 105 \beta_1) q^{67} + (3 \beta_{3} + 60) q^{69} + ( - 29 \beta_{3} - 554) q^{71} + ( - 6 \beta_{2} + 246 \beta_1) q^{73} + ( - 5 \beta_{2} - 1300 \beta_1) q^{77} + ( - 28 \beta_{3} - 664) q^{79} + 81 q^{81} + ( - 32 \beta_{2} + 14 \beta_1) q^{83} + (3 \beta_{2} + 360 \beta_1) q^{87} + ( - 4 \beta_{3} - 712) q^{89} + (43 \beta_{3} - 913) q^{91} + ( - 27 \beta_{2} - 327 \beta_1) q^{93} + (16 \beta_{2} - 1185 \beta_1) q^{97} + ( - 27 \beta_{3} - 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} + 32 q^{11} - 420 q^{19} - 12 q^{21} + 480 q^{29} - 436 q^{31} - 492 q^{39} + 168 q^{41} - 376 q^{49} + 48 q^{51} - 2168 q^{59} + 1548 q^{61} + 240 q^{69} - 2216 q^{71} - 2656 q^{79} + 324 q^{81} - 2848 q^{89} - 3652 q^{91} - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 55x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 28\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 164\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 110 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 110 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -7\beta_{2} + 41\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
5.72015i
4.72015i
4.72015i
5.72015i
0 3.00000i 0 0 0 21.8806i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 19.8806i 0 −9.00000 0
49.3 0 3.00000i 0 0 0 19.8806i 0 −9.00000 0
49.4 0 3.00000i 0 0 0 21.8806i 0 −9.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.f.x 4
4.b odd 2 1 600.4.f.j 4
5.b even 2 1 inner 1200.4.f.x 4
5.c odd 4 1 1200.4.a.bp 2
5.c odd 4 1 1200.4.a.br 2
12.b even 2 1 1800.4.f.z 4
20.d odd 2 1 600.4.f.j 4
20.e even 4 1 600.4.a.s 2
20.e even 4 1 600.4.a.u yes 2
60.h even 2 1 1800.4.f.z 4
60.l odd 4 1 1800.4.a.bm 2
60.l odd 4 1 1800.4.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.s 2 20.e even 4 1
600.4.a.u yes 2 20.e even 4 1
600.4.f.j 4 4.b odd 2 1
600.4.f.j 4 20.d odd 2 1
1200.4.a.bp 2 5.c odd 4 1
1200.4.a.br 2 5.c odd 4 1
1200.4.f.x 4 1.a even 1 1 trivial
1200.4.f.x 4 5.b even 2 1 inner
1800.4.a.bm 2 60.l odd 4 1
1800.4.a.bo 2 60.l odd 4 1
1800.4.f.z 4 12.b even 2 1
1800.4.f.z 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} + 874T_{7}^{2} + 189225 \) Copy content Toggle raw display
\( T_{11}^{2} - 16T_{11} - 3860 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 874 T^{2} + 189225 \) Copy content Toggle raw display
$11$ \( (T^{2} - 16 T - 3860)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 6850 T^{2} + 3969 \) Copy content Toggle raw display
$17$ \( T^{4} + 21832 T^{2} + 118461456 \) Copy content Toggle raw display
$19$ \( (T^{2} + 210 T + 10589)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1672 T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 240 T + 13964)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 218 T - 23435)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 153448 T^{2} + 109746576 \) Copy content Toggle raw display
$41$ \( (T^{2} - 84 T - 2160)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 59338 T^{2} + 61921161 \) Copy content Toggle raw display
$47$ \( T^{4} + 262856 T^{2} + 34339600 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 2057529600 \) Copy content Toggle raw display
$59$ \( (T^{2} + 1084 T + 265860)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 774 T + 142793)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 343230767881 \) Copy content Toggle raw display
$71$ \( (T^{2} + 1108 T - 59760)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 2008832400 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1328 T + 99072)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 199155127824 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1424 T + 499968)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 1670838026881 \) Copy content Toggle raw display
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