Properties

Label 900.6.a.w
Level $900$
Weight $6$
Character orbit 900.a
Self dual yes
Analytic conductor $144.345$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(144.345437832\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.535753.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 186x - 432 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 60)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 29) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 29) q^{7} + (4 \beta_{2} + \beta_1 - 48) q^{11} + 3 \beta_1 q^{13} + (2 \beta_{2} + 7 \beta_1 + 290) q^{17} + (6 \beta_{2} + 6 \beta_1 + 1002) q^{19} + (14 \beta_{2} - 2 \beta_1 - 346) q^{23} + ( - 3 \beta_{2} + 6 \beta_1 - 2655) q^{29} + (42 \beta_{2} - 12 \beta_1 + 58) q^{31} + (40 \beta_{2} + 21 \beta_1 - 3968) q^{37} + ( - 2 \beta_{2} - 32 \beta_1 - 2724) q^{41} + ( - 8 \beta_{2} - 72 \beta_1 - 8732) q^{43} + (100 \beta_{2} - 46 \beta_1 + 12664) q^{47} + ( - 102 \beta_{2} + 60 \beta_1 + 4575) q^{49} + (30 \beta_{2} - 15 \beta_1 + 3702) q^{53} + ( - 4 \beta_{2} + 89 \beta_1 + 15408) q^{59} + ( - 168 \beta_{2} - 168 \beta_1 + 986) q^{61} + (158 \beta_{2} + 48 \beta_1 - 42058) q^{67} + ( - 12 \beta_{2} - 66 \beta_1 + 26796) q^{71} + ( - 212 \beta_{2} + 150 \beta_1 - 14372) q^{73} + ( - 200 \beta_{2} + 254 \beta_1 + 78136) q^{77} + (126 \beta_{2} + 180 \beta_1 - 21450) q^{79} + ( - 316 \beta_{2} - 92 \beta_1 + 41984) q^{83} + ( - 284 \beta_{2} - 188 \beta_1 + 12582) q^{89} + (420 \beta_{2} + 42 \beta_1 - 16260) q^{91} + ( - 340 \beta_{2} - 84 \beta_1 - 110452) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 88 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 88 q^{7} - 148 q^{11} + 868 q^{17} + 3000 q^{19} - 1052 q^{23} - 7962 q^{29} + 132 q^{31} - 11944 q^{37} - 8170 q^{41} - 26188 q^{43} + 37892 q^{47} + 13827 q^{49} + 11076 q^{53} + 46228 q^{59} + 3126 q^{61} - 126332 q^{67} + 80400 q^{71} - 42904 q^{73} + 234608 q^{77} - 64476 q^{79} + 126268 q^{83} + 38030 q^{89} - 49200 q^{91} - 331016 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 186x - 432 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{2} + 65\nu + 600 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{2} - 5\nu - 621 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 7 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 13\beta_{2} + \beta _1 + 2491 ) / 20 \) 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
−2.43168
−11.7229
15.1546
0 0 0 0 0 −222.092 0 0 0
1.2 0 0 0 0 0 12.5817 0 0 0
1.3 0 0 0 0 0 121.510 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.6.a.w 3
3.b odd 2 1 300.6.a.i 3
5.b even 2 1 900.6.a.x 3
5.c odd 4 2 180.6.d.d 6
15.d odd 2 1 300.6.a.j 3
15.e even 4 2 60.6.d.a 6
20.e even 4 2 720.6.f.m 6
60.l odd 4 2 240.6.f.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.d.a 6 15.e even 4 2
180.6.d.d 6 5.c odd 4 2
240.6.f.d 6 60.l odd 4 2
300.6.a.i 3 3.b odd 2 1
300.6.a.j 3 15.d odd 2 1
720.6.f.m 6 20.e even 4 2
900.6.a.w 3 1.a even 1 1 trivial
900.6.a.x 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(900))\):

\( T_{7}^{3} + 88T_{7}^{2} - 28252T_{7} + 339536 \) Copy content Toggle raw display
\( T_{11}^{3} + 148T_{11}^{2} - 480532T_{11} - 78696304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 88 T^{2} - 28252 T + 339536 \) Copy content Toggle raw display
$11$ \( T^{3} + 148 T^{2} + \cdots - 78696304 \) Copy content Toggle raw display
$13$ \( T^{3} - 540900 T + 152928000 \) Copy content Toggle raw display
$17$ \( T^{3} - 868 T^{2} + \cdots + 2242315984 \) Copy content Toggle raw display
$19$ \( T^{3} - 3000 T^{2} + \cdots + 829440000 \) Copy content Toggle raw display
$23$ \( T^{3} + 1052 T^{2} + \cdots + 3920087104 \) Copy content Toggle raw display
$29$ \( T^{3} + 7962 T^{2} + \cdots + 12509144064 \) Copy content Toggle raw display
$31$ \( T^{3} - 132 T^{2} + \cdots + 233735426816 \) Copy content Toggle raw display
$37$ \( T^{3} + 11944 T^{2} + \cdots - 371106407008 \) Copy content Toggle raw display
$41$ \( T^{3} + 8170 T^{2} + \cdots - 323508673000 \) Copy content Toggle raw display
$43$ \( T^{3} + 26188 T^{2} + \cdots - 3946674940864 \) Copy content Toggle raw display
$47$ \( T^{3} - 37892 T^{2} + \cdots + 8134093677056 \) Copy content Toggle raw display
$53$ \( T^{3} - 11076 T^{2} + \cdots + 230385380112 \) Copy content Toggle raw display
$59$ \( T^{3} - 46228 T^{2} + \cdots + 7848174247024 \) Copy content Toggle raw display
$61$ \( T^{3} - 3126 T^{2} + \cdots + 20932589363512 \) Copy content Toggle raw display
$67$ \( T^{3} + 126332 T^{2} + \cdots + 36547845885184 \) Copy content Toggle raw display
$71$ \( T^{3} - 80400 T^{2} + \cdots - 13864753152000 \) Copy content Toggle raw display
$73$ \( T^{3} + 42904 T^{2} + \cdots - 75705138064768 \) Copy content Toggle raw display
$79$ \( T^{3} + 64476 T^{2} + \cdots - 33510082240512 \) Copy content Toggle raw display
$83$ \( T^{3} - 126268 T^{2} + \cdots + 94076318344384 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 129133988519000 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 987599138549248 \) Copy content Toggle raw display
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