Properties

Label 900.4.a.s
Level $900$
Weight $4$
Character orbit 900.a
Self dual yes
Analytic conductor $53.102$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,4,Mod(1,900)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.1");
 
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.a (trivial)

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: yes
Analytic conductor: \(53.1017190052\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 2\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{7} - 20 q^{11} - 6 \beta q^{13} + 8 \beta q^{17} + 84 q^{19} - 7 \beta q^{23} + 6 q^{29} - 224 q^{31} - 14 \beta q^{37} - 266 q^{41} - 35 \beta q^{43} + 43 \beta q^{47} - 267 q^{49} + 42 \beta q^{53} - 28 q^{59} + 182 q^{61} - 49 \beta q^{67} - 408 q^{71} + 124 \beta q^{73} - 20 \beta q^{77} - 48 q^{79} + 23 \beta q^{83} - 1526 q^{89} - 456 q^{91} - 64 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 40 q^{11} + 168 q^{19} + 12 q^{29} - 448 q^{31} - 532 q^{41} - 534 q^{49} - 56 q^{59} + 364 q^{61} - 816 q^{71} - 96 q^{79} - 3052 q^{89} - 912 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.35890
4.35890
0 0 0 0 0 −8.71780 0 0 0
1.2 0 0 0 0 0 8.71780 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

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.a.s 2
3.b odd 2 1 100.4.a.d 2
5.b even 2 1 inner 900.4.a.s 2
5.c odd 4 2 180.4.d.a 2
12.b even 2 1 400.4.a.w 2
15.d odd 2 1 100.4.a.d 2
15.e even 4 2 20.4.c.a 2
20.e even 4 2 720.4.f.a 2
24.f even 2 1 1600.4.a.ck 2
24.h odd 2 1 1600.4.a.cj 2
60.h even 2 1 400.4.a.w 2
60.l odd 4 2 80.4.c.b 2
105.k odd 4 2 980.4.e.a 2
120.i odd 2 1 1600.4.a.cj 2
120.m even 2 1 1600.4.a.ck 2
120.q odd 4 2 320.4.c.b 2
120.w even 4 2 320.4.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.c.a 2 15.e even 4 2
80.4.c.b 2 60.l odd 4 2
100.4.a.d 2 3.b odd 2 1
100.4.a.d 2 15.d odd 2 1
180.4.d.a 2 5.c odd 4 2
320.4.c.a 2 120.w even 4 2
320.4.c.b 2 120.q odd 4 2
400.4.a.w 2 12.b even 2 1
400.4.a.w 2 60.h even 2 1
720.4.f.a 2 20.e even 4 2
900.4.a.s 2 1.a even 1 1 trivial
900.4.a.s 2 5.b even 2 1 inner
980.4.e.a 2 105.k odd 4 2
1600.4.a.cj 2 24.h odd 2 1
1600.4.a.cj 2 120.i odd 2 1
1600.4.a.ck 2 24.f even 2 1
1600.4.a.ck 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(900))\):

\( T_{7}^{2} - 76 \) Copy content Toggle raw display
\( T_{11} + 20 \) 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
$7$ \( T^{2} - 76 \) Copy content Toggle raw display
$11$ \( (T + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2736 \) Copy content Toggle raw display
$17$ \( T^{2} - 4864 \) Copy content Toggle raw display
$19$ \( (T - 84)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3724 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 14896 \) Copy content Toggle raw display
$41$ \( (T + 266)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 93100 \) Copy content Toggle raw display
$47$ \( T^{2} - 140524 \) Copy content Toggle raw display
$53$ \( T^{2} - 134064 \) Copy content Toggle raw display
$59$ \( (T + 28)^{2} \) Copy content Toggle raw display
$61$ \( (T - 182)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 182476 \) Copy content Toggle raw display
$71$ \( (T + 408)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1168576 \) Copy content Toggle raw display
$79$ \( (T + 48)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 40204 \) Copy content Toggle raw display
$89$ \( (T + 1526)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 311296 \) Copy content Toggle raw display
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