Properties

Label 900.4.d.c
Level $900$
Weight $4$
Character orbit 900.d
Analytic conductor $53.102$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,4,Mod(649,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("900.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.d (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: \(53.1017190052\)
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 12)
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.

\(f(q)\) \(=\) \( q + 4 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta q^{7} - 36 q^{11} + 5 \beta q^{13} - 9 \beta q^{17} + 100 q^{19} + 36 \beta q^{23} - 234 q^{29} - 16 q^{31} - 113 \beta q^{37} - 90 q^{41} - 226 \beta q^{43} - 216 \beta q^{47} + 279 q^{49} + 207 \beta q^{53} - 684 q^{59} + 422 q^{61} + 166 \beta q^{67} + 360 q^{71} - 13 \beta q^{73} - 144 \beta q^{77} - 512 q^{79} - 594 \beta q^{83} - 630 q^{89} - 80 q^{91} - 527 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 72 q^{11} + 200 q^{19} - 468 q^{29} - 32 q^{31} - 180 q^{41} + 558 q^{49} - 1368 q^{59} + 844 q^{61} + 720 q^{71} - 1024 q^{79} - 1260 q^{89} - 160 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 8.00000i 0 0 0
649.2 0 0 0 0 0 8.00000i 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 900.4.d.c 2
3.b odd 2 1 300.4.d.e 2
5.b even 2 1 inner 900.4.d.c 2
5.c odd 4 1 36.4.a.a 1
5.c odd 4 1 900.4.a.g 1
12.b even 2 1 1200.4.f.d 2
15.d odd 2 1 300.4.d.e 2
15.e even 4 1 12.4.a.a 1
15.e even 4 1 300.4.a.b 1
20.e even 4 1 144.4.a.g 1
35.f even 4 1 1764.4.a.b 1
35.k even 12 2 1764.4.k.o 2
35.l odd 12 2 1764.4.k.b 2
40.i odd 4 1 576.4.a.b 1
40.k even 4 1 576.4.a.a 1
45.k odd 12 2 324.4.e.a 2
45.l even 12 2 324.4.e.h 2
60.h even 2 1 1200.4.f.d 2
60.l odd 4 1 48.4.a.a 1
60.l odd 4 1 1200.4.a.be 1
105.k odd 4 1 588.4.a.c 1
105.w odd 12 2 588.4.i.e 2
105.x even 12 2 588.4.i.d 2
120.q odd 4 1 192.4.a.l 1
120.w even 4 1 192.4.a.f 1
165.l odd 4 1 1452.4.a.d 1
195.j odd 4 1 2028.4.b.c 2
195.s even 4 1 2028.4.a.c 1
195.u odd 4 1 2028.4.b.c 2
240.z odd 4 1 768.4.d.j 2
240.bb even 4 1 768.4.d.g 2
240.bd odd 4 1 768.4.d.j 2
240.bf even 4 1 768.4.d.g 2
420.w even 4 1 2352.4.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 15.e even 4 1
36.4.a.a 1 5.c odd 4 1
48.4.a.a 1 60.l odd 4 1
144.4.a.g 1 20.e even 4 1
192.4.a.f 1 120.w even 4 1
192.4.a.l 1 120.q odd 4 1
300.4.a.b 1 15.e even 4 1
300.4.d.e 2 3.b odd 2 1
300.4.d.e 2 15.d odd 2 1
324.4.e.a 2 45.k odd 12 2
324.4.e.h 2 45.l even 12 2
576.4.a.a 1 40.k even 4 1
576.4.a.b 1 40.i odd 4 1
588.4.a.c 1 105.k odd 4 1
588.4.i.d 2 105.x even 12 2
588.4.i.e 2 105.w odd 12 2
768.4.d.g 2 240.bb even 4 1
768.4.d.g 2 240.bf even 4 1
768.4.d.j 2 240.z odd 4 1
768.4.d.j 2 240.bd odd 4 1
900.4.a.g 1 5.c odd 4 1
900.4.d.c 2 1.a even 1 1 trivial
900.4.d.c 2 5.b even 2 1 inner
1200.4.a.be 1 60.l odd 4 1
1200.4.f.d 2 12.b even 2 1
1200.4.f.d 2 60.h even 2 1
1452.4.a.d 1 165.l odd 4 1
1764.4.a.b 1 35.f even 4 1
1764.4.k.b 2 35.l odd 12 2
1764.4.k.o 2 35.k even 12 2
2028.4.a.c 1 195.s even 4 1
2028.4.b.c 2 195.j odd 4 1
2028.4.b.c 2 195.u odd 4 1
2352.4.a.bk 1 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11} + 36 \) 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
$7$ \( T^{2} + 64 \) Copy content Toggle raw display
$11$ \( (T + 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 100 \) Copy content Toggle raw display
$17$ \( T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T - 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5184 \) Copy content Toggle raw display
$29$ \( (T + 234)^{2} \) Copy content Toggle raw display
$31$ \( (T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 51076 \) Copy content Toggle raw display
$41$ \( (T + 90)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 204304 \) Copy content Toggle raw display
$47$ \( T^{2} + 186624 \) Copy content Toggle raw display
$53$ \( T^{2} + 171396 \) Copy content Toggle raw display
$59$ \( (T + 684)^{2} \) Copy content Toggle raw display
$61$ \( (T - 422)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 110224 \) Copy content Toggle raw display
$71$ \( (T - 360)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 676 \) Copy content Toggle raw display
$79$ \( (T + 512)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1411344 \) Copy content Toggle raw display
$89$ \( (T + 630)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1110916 \) Copy content Toggle raw display
show more
show less