Properties

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

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Show commands: Magma / PariGP / SageMath

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 120)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 4 i q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} + 4 i q^{7} - 9 q^{9} - 72 q^{11} - 6 i q^{13} - 38 i q^{17} + 52 q^{19} - 12 q^{21} - 152 i q^{23} - 27 i q^{27} + 78 q^{29} - 120 q^{31} - 216 i q^{33} + 150 i q^{37} + 18 q^{39} + 362 q^{41} + 484 i q^{43} + 280 i q^{47} + 327 q^{49} + 114 q^{51} - 670 i q^{53} + 156 i q^{57} + 696 q^{59} + 222 q^{61} - 36 i q^{63} - 4 i q^{67} + 456 q^{69} - 96 q^{71} + 178 i q^{73} - 288 i q^{77} - 632 q^{79} + 81 q^{81} + 612 i q^{83} + 234 i q^{87} - 994 q^{89} + 24 q^{91} - 360 i q^{93} - 1634 i q^{97} + 648 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 144 q^{11} + 104 q^{19} - 24 q^{21} + 156 q^{29} - 240 q^{31} + 36 q^{39} + 724 q^{41} + 654 q^{49} + 228 q^{51} + 1392 q^{59} + 444 q^{61} + 912 q^{69} - 192 q^{71} - 1264 q^{79} + 162 q^{81} - 1988 q^{89} + 48 q^{91} + 1296 q^{99}+O(q^{100}) \) 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 4.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 4.00000i 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.a 2
4.b odd 2 1 600.4.f.i 2
5.b even 2 1 inner 1200.4.f.a 2
5.c odd 4 1 240.4.a.h 1
5.c odd 4 1 1200.4.a.k 1
12.b even 2 1 1800.4.f.a 2
15.e even 4 1 720.4.a.v 1
20.d odd 2 1 600.4.f.i 2
20.e even 4 1 120.4.a.a 1
20.e even 4 1 600.4.a.l 1
40.i odd 4 1 960.4.a.o 1
40.k even 4 1 960.4.a.bf 1
60.h even 2 1 1800.4.f.a 2
60.l odd 4 1 360.4.a.l 1
60.l odd 4 1 1800.4.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.a 1 20.e even 4 1
240.4.a.h 1 5.c odd 4 1
360.4.a.l 1 60.l odd 4 1
600.4.a.l 1 20.e even 4 1
600.4.f.i 2 4.b odd 2 1
600.4.f.i 2 20.d odd 2 1
720.4.a.v 1 15.e even 4 1
960.4.a.o 1 40.i odd 4 1
960.4.a.bf 1 40.k even 4 1
1200.4.a.k 1 5.c odd 4 1
1200.4.f.a 2 1.a even 1 1 trivial
1200.4.f.a 2 5.b even 2 1 inner
1800.4.a.n 1 60.l odd 4 1
1800.4.f.a 2 12.b even 2 1
1800.4.f.a 2 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 \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 72 \) 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
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 72)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 1444 \) Copy content Toggle raw display
$19$ \( (T - 52)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 23104 \) Copy content Toggle raw display
$29$ \( (T - 78)^{2} \) Copy content Toggle raw display
$31$ \( (T + 120)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 22500 \) Copy content Toggle raw display
$41$ \( (T - 362)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 234256 \) Copy content Toggle raw display
$47$ \( T^{2} + 78400 \) Copy content Toggle raw display
$53$ \( T^{2} + 448900 \) Copy content Toggle raw display
$59$ \( (T - 696)^{2} \) Copy content Toggle raw display
$61$ \( (T - 222)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 31684 \) Copy content Toggle raw display
$79$ \( (T + 632)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 374544 \) Copy content Toggle raw display
$89$ \( (T + 994)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2669956 \) Copy content Toggle raw display
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