Properties

Label 450.4.a.r
Level $450$
Weight $4$
Character orbit 450.a
Self dual yes
Analytic conductor $26.551$
Analytic rank $0$
Dimension $1$
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] = [450,4,Mod(1,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("450.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.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: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 4 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 4 q^{7} + 8 q^{8} + 48 q^{11} - 2 q^{13} + 8 q^{14} + 16 q^{16} - 114 q^{17} + 140 q^{19} + 96 q^{22} + 72 q^{23} - 4 q^{26} + 16 q^{28} - 210 q^{29} + 272 q^{31} + 32 q^{32} - 228 q^{34} + 334 q^{37} + 280 q^{38} + 198 q^{41} + 268 q^{43} + 192 q^{44} + 144 q^{46} + 216 q^{47} - 327 q^{49} - 8 q^{52} - 78 q^{53} + 32 q^{56} - 420 q^{58} - 240 q^{59} + 302 q^{61} + 544 q^{62} + 64 q^{64} - 596 q^{67} - 456 q^{68} + 768 q^{71} + 478 q^{73} + 668 q^{74} + 560 q^{76} + 192 q^{77} - 640 q^{79} + 396 q^{82} - 348 q^{83} + 536 q^{86} + 384 q^{88} - 210 q^{89} - 8 q^{91} + 288 q^{92} + 432 q^{94} + 1534 q^{97} - 654 q^{98}+O(q^{100}) \) 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
0
2.00000 0 4.00000 0 0 4.00000 8.00000 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 450.4.a.r 1
3.b odd 2 1 150.4.a.b 1
5.b even 2 1 90.4.a.c 1
5.c odd 4 2 450.4.c.j 2
12.b even 2 1 1200.4.a.ba 1
15.d odd 2 1 30.4.a.b 1
15.e even 4 2 150.4.c.c 2
20.d odd 2 1 720.4.a.y 1
45.h odd 6 2 810.4.e.i 2
45.j even 6 2 810.4.e.p 2
60.h even 2 1 240.4.a.b 1
60.l odd 4 2 1200.4.f.r 2
105.g even 2 1 1470.4.a.r 1
120.i odd 2 1 960.4.a.n 1
120.m even 2 1 960.4.a.bg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.b 1 15.d odd 2 1
90.4.a.c 1 5.b even 2 1
150.4.a.b 1 3.b odd 2 1
150.4.c.c 2 15.e even 4 2
240.4.a.b 1 60.h even 2 1
450.4.a.r 1 1.a even 1 1 trivial
450.4.c.j 2 5.c odd 4 2
720.4.a.y 1 20.d odd 2 1
810.4.e.i 2 45.h odd 6 2
810.4.e.p 2 45.j even 6 2
960.4.a.n 1 120.i odd 2 1
960.4.a.bg 1 120.m even 2 1
1200.4.a.ba 1 12.b even 2 1
1200.4.f.r 2 60.l odd 4 2
1470.4.a.r 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} - 48 \) Copy content Toggle raw display
\( T_{17} + 114 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T - 48 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 114 \) Copy content Toggle raw display
$19$ \( T - 140 \) Copy content Toggle raw display
$23$ \( T - 72 \) Copy content Toggle raw display
$29$ \( T + 210 \) Copy content Toggle raw display
$31$ \( T - 272 \) Copy content Toggle raw display
$37$ \( T - 334 \) Copy content Toggle raw display
$41$ \( T - 198 \) Copy content Toggle raw display
$43$ \( T - 268 \) Copy content Toggle raw display
$47$ \( T - 216 \) Copy content Toggle raw display
$53$ \( T + 78 \) Copy content Toggle raw display
$59$ \( T + 240 \) Copy content Toggle raw display
$61$ \( T - 302 \) Copy content Toggle raw display
$67$ \( T + 596 \) Copy content Toggle raw display
$71$ \( T - 768 \) Copy content Toggle raw display
$73$ \( T - 478 \) Copy content Toggle raw display
$79$ \( T + 640 \) Copy content Toggle raw display
$83$ \( T + 348 \) Copy content Toggle raw display
$89$ \( T + 210 \) Copy content Toggle raw display
$97$ \( T - 1534 \) Copy content Toggle raw display
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