Properties

Label 1200.4.f.k
Level $1200$
Weight $4$
Character orbit 1200.f
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 i q^{3} + 13 i q^{7} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 i q^{3} + 13 i q^{7} - 9 q^{9} - 6 q^{11} - 5 i q^{13} - 78 i q^{17} + 65 q^{19} + 39 q^{21} + 138 i q^{23} + 27 i q^{27} - 66 q^{29} - 299 q^{31} + 18 i q^{33} - 214 i q^{37} - 15 q^{39} + 360 q^{41} + 203 i q^{43} - 78 i q^{47} + 174 q^{49} - 234 q^{51} - 636 i q^{53} - 195 i q^{57} + 786 q^{59} + 467 q^{61} - 117 i q^{63} + 217 i q^{67} + 414 q^{69} + 360 q^{71} + 286 i q^{73} - 78 i q^{77} + 272 q^{79} + 81 q^{81} + 498 i q^{83} + 198 i q^{87} + 65 q^{91} + 897 i q^{93} - 511 i q^{97} + 54 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 12 q^{11} + 130 q^{19} + 78 q^{21} - 132 q^{29} - 598 q^{31} - 30 q^{39} + 720 q^{41} + 348 q^{49} - 468 q^{51} + 1572 q^{59} + 934 q^{61} + 828 q^{69} + 720 q^{71} + 544 q^{79} + 162 q^{81} + 130 q^{91} + 108 q^{99}+O(q^{100}) \) 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
1.00000i
1.00000i
0 3.00000i 0 0 0 13.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 13.0000i 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.k 2
4.b odd 2 1 300.4.d.c 2
5.b even 2 1 inner 1200.4.f.k 2
5.c odd 4 1 1200.4.a.d 1
5.c odd 4 1 1200.4.a.bi 1
12.b even 2 1 900.4.d.e 2
20.d odd 2 1 300.4.d.c 2
20.e even 4 1 300.4.a.a 1
20.e even 4 1 300.4.a.h yes 1
60.h even 2 1 900.4.d.e 2
60.l odd 4 1 900.4.a.f 1
60.l odd 4 1 900.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.4.a.a 1 20.e even 4 1
300.4.a.h yes 1 20.e even 4 1
300.4.d.c 2 4.b odd 2 1
300.4.d.c 2 20.d odd 2 1
900.4.a.f 1 60.l odd 4 1
900.4.a.l 1 60.l odd 4 1
900.4.d.e 2 12.b even 2 1
900.4.d.e 2 60.h even 2 1
1200.4.a.d 1 5.c odd 4 1
1200.4.a.bi 1 5.c odd 4 1
1200.4.f.k 2 1.a even 1 1 trivial
1200.4.f.k 2 5.b even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} + 169 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 169 \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} + 6084 \) Copy content Toggle raw display
$19$ \( (T - 65)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 19044 \) Copy content Toggle raw display
$29$ \( (T + 66)^{2} \) Copy content Toggle raw display
$31$ \( (T + 299)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 45796 \) Copy content Toggle raw display
$41$ \( (T - 360)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 41209 \) Copy content Toggle raw display
$47$ \( T^{2} + 6084 \) Copy content Toggle raw display
$53$ \( T^{2} + 404496 \) Copy content Toggle raw display
$59$ \( (T - 786)^{2} \) Copy content Toggle raw display
$61$ \( (T - 467)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 47089 \) Copy content Toggle raw display
$71$ \( (T - 360)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 81796 \) Copy content Toggle raw display
$79$ \( (T - 272)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 248004 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 261121 \) Copy content Toggle raw display
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