Properties

Label 1800.4.f.x
Level $1800$
Weight $4$
Character orbit 1800.f
Analytic conductor $106.203$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,4,Mod(649,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, 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("1800.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.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: \(106.203438010\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 72)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 6 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 \beta q^{7} + 64 q^{11} + 29 \beta q^{13} - 16 \beta q^{17} + 136 q^{19} - 64 \beta q^{23} + 144 q^{29} + 20 q^{31} + 9 \beta q^{37} - 288 q^{41} - 100 \beta q^{43} - 192 \beta q^{47} + 199 q^{49} + 248 \beta q^{53} + 128 q^{59} - 458 q^{61} + 248 \beta q^{67} + 512 q^{71} - 301 \beta q^{73} + 384 \beta q^{77} - 1108 q^{79} + 352 \beta q^{83} + 960 q^{89} - 696 q^{91} - 103 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 128 q^{11} + 272 q^{19} + 288 q^{29} + 40 q^{31} - 576 q^{41} + 398 q^{49} + 256 q^{59} - 916 q^{61} + 1024 q^{71} - 2216 q^{79} + 1920 q^{89} - 1392 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 0 0 12.0000i 0 0 0
649.2 0 0 0 0 0 12.0000i 0 0 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 1800.4.f.x 2
3.b odd 2 1 1800.4.f.b 2
5.b even 2 1 inner 1800.4.f.x 2
5.c odd 4 1 72.4.a.d yes 1
5.c odd 4 1 1800.4.a.ba 1
15.d odd 2 1 1800.4.f.b 2
15.e even 4 1 72.4.a.a 1
15.e even 4 1 1800.4.a.z 1
20.e even 4 1 144.4.a.f 1
40.i odd 4 1 576.4.a.c 1
40.k even 4 1 576.4.a.d 1
45.k odd 12 2 648.4.i.a 2
45.l even 12 2 648.4.i.l 2
60.l odd 4 1 144.4.a.a 1
120.q odd 4 1 576.4.a.x 1
120.w even 4 1 576.4.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 15.e even 4 1
72.4.a.d yes 1 5.c odd 4 1
144.4.a.a 1 60.l odd 4 1
144.4.a.f 1 20.e even 4 1
576.4.a.c 1 40.i odd 4 1
576.4.a.d 1 40.k even 4 1
576.4.a.w 1 120.w even 4 1
576.4.a.x 1 120.q odd 4 1
648.4.i.a 2 45.k odd 12 2
648.4.i.l 2 45.l even 12 2
1800.4.a.z 1 15.e even 4 1
1800.4.a.ba 1 5.c odd 4 1
1800.4.f.b 2 3.b odd 2 1
1800.4.f.b 2 15.d odd 2 1
1800.4.f.x 2 1.a even 1 1 trivial
1800.4.f.x 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 \(S_{4}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{11} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 144 \) Copy content Toggle raw display
$11$ \( (T - 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3364 \) Copy content Toggle raw display
$17$ \( T^{2} + 1024 \) Copy content Toggle raw display
$19$ \( (T - 136)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16384 \) Copy content Toggle raw display
$29$ \( (T - 144)^{2} \) Copy content Toggle raw display
$31$ \( (T - 20)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 324 \) Copy content Toggle raw display
$41$ \( (T + 288)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 40000 \) Copy content Toggle raw display
$47$ \( T^{2} + 147456 \) Copy content Toggle raw display
$53$ \( T^{2} + 246016 \) Copy content Toggle raw display
$59$ \( (T - 128)^{2} \) Copy content Toggle raw display
$61$ \( (T + 458)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 246016 \) Copy content Toggle raw display
$71$ \( (T - 512)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 362404 \) Copy content Toggle raw display
$79$ \( (T + 1108)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 495616 \) Copy content Toggle raw display
$89$ \( (T - 960)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 42436 \) Copy content Toggle raw display
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