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Properties

Label 1200.4.f.f
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

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1200,4,Mod(49,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-18,0,-44,0,0,0,0,0,0,0,-106] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content 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 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 19 i q^{7} - 9 q^{9} - 22 q^{11} - i q^{13} - 58 i q^{17} - 53 q^{19} - 57 q^{21} + 58 i q^{23} - 27 i q^{27} - 22 q^{29} + 35 q^{31} - 66 i q^{33} - 270 i q^{37} + 3 q^{39} - 468 q^{41} + \cdots + 198 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} - 44 q^{11} - 106 q^{19} - 114 q^{21} - 44 q^{29} + 70 q^{31} + 6 q^{39} - 936 q^{41} - 36 q^{49} + 348 q^{51} + 892 q^{59} + 254 q^{61} - 348 q^{69} - 72 q^{71} + 2736 q^{79} + 162 q^{81}+ \cdots + 396 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 − 19.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 19.0000i 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.f 2
4.b odd 2 1 600.4.f.h 2
5.b even 2 1 inner 1200.4.f.f 2
5.c odd 4 1 1200.4.a.n 1
5.c odd 4 1 1200.4.a.x 1
12.b even 2 1 1800.4.f.g 2
20.d odd 2 1 600.4.f.h 2
20.e even 4 1 600.4.a.g ✓ 1
20.e even 4 1 600.4.a.j yes 1
60.h even 2 1 1800.4.f.g 2
60.l odd 4 1 1800.4.a.g 1
60.l odd 4 1 1800.4.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.g ✓ 1 20.e even 4 1
600.4.a.j yes 1 20.e even 4 1
600.4.f.h 2 4.b odd 2 1
600.4.f.h 2 20.d odd 2 1
1200.4.a.n 1 5.c odd 4 1
1200.4.a.x 1 5.c odd 4 1
1200.4.f.f 2 1.a even 1 1 trivial
1200.4.f.f 2 5.b even 2 1 inner
1800.4.a.g 1 60.l odd 4 1
1800.4.a.bf 1 60.l odd 4 1
1800.4.f.g 2 12.b even 2 1
1800.4.f.g 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 \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 361 \) Copy content Toggle raw display
\( T_{11} + 22 \) 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
$7$ \( T^{2} + 361 \) Copy content Toggle raw display
$11$ \( (T + 22)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3364 \) Copy content Toggle raw display
$19$ \( (T + 53)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3364 \) Copy content Toggle raw display
$29$ \( (T + 22)^{2} \) Copy content Toggle raw display
$31$ \( (T - 35)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 72900 \) Copy content Toggle raw display
$41$ \( (T + 468)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 185761 \) Copy content Toggle raw display
$47$ \( T^{2} + 52900 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T - 446)^{2} \) Copy content Toggle raw display
$61$ \( (T - 127)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 657721 \) Copy content Toggle raw display
$71$ \( (T + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 272484 \) Copy content Toggle raw display
$79$ \( (T - 1368)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1295044 \) Copy content Toggle raw display
$89$ \( (T + 144)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1164241 \) Copy content Toggle raw display
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