Properties

Label 450.4.a.p
Level $450$
Weight $4$
Character orbit 450.a
Self dual yes
Analytic conductor $26.551$
Analytic rank $1$
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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 2 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 2 q^{7} + 8 q^{8} - 70 q^{11} - 54 q^{13} + 4 q^{14} + 16 q^{16} - 22 q^{17} + 24 q^{19} - 140 q^{22} - 100 q^{23} - 108 q^{26} + 8 q^{28} - 216 q^{29} + 208 q^{31} + 32 q^{32} - 44 q^{34} + 254 q^{37} + 48 q^{38} + 206 q^{41} - 292 q^{43} - 280 q^{44} - 200 q^{46} - 320 q^{47} - 339 q^{49} - 216 q^{52} - 402 q^{53} + 16 q^{56} - 432 q^{58} + 370 q^{59} - 550 q^{61} + 416 q^{62} + 64 q^{64} - 728 q^{67} - 88 q^{68} + 540 q^{71} - 604 q^{73} + 508 q^{74} + 96 q^{76} - 140 q^{77} + 792 q^{79} + 412 q^{82} + 404 q^{83} - 584 q^{86} - 560 q^{88} + 938 q^{89} - 108 q^{91} - 400 q^{92} - 640 q^{94} - 56 q^{97} - 678 q^{98}+O(q^{100}) \) 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 2.00000 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.p 1
3.b odd 2 1 150.4.a.d 1
5.b even 2 1 450.4.a.e 1
5.c odd 4 2 90.4.c.a 2
12.b even 2 1 1200.4.a.h 1
15.d odd 2 1 150.4.a.f 1
15.e even 4 2 30.4.c.a 2
20.e even 4 2 720.4.f.c 2
60.h even 2 1 1200.4.a.bc 1
60.l odd 4 2 240.4.f.d 2
120.q odd 4 2 960.4.f.d 2
120.w even 4 2 960.4.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.c.a 2 15.e even 4 2
90.4.c.a 2 5.c odd 4 2
150.4.a.d 1 3.b odd 2 1
150.4.a.f 1 15.d odd 2 1
240.4.f.d 2 60.l odd 4 2
450.4.a.e 1 5.b even 2 1
450.4.a.p 1 1.a even 1 1 trivial
720.4.f.c 2 20.e even 4 2
960.4.f.c 2 120.w even 4 2
960.4.f.d 2 120.q odd 4 2
1200.4.a.h 1 12.b even 2 1
1200.4.a.bc 1 60.h even 2 1

Hecke kernels

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

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} + 70 \) Copy content Toggle raw display
\( T_{17} + 22 \) 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
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 70 \) Copy content Toggle raw display
$13$ \( T + 54 \) Copy content Toggle raw display
$17$ \( T + 22 \) Copy content Toggle raw display
$19$ \( T - 24 \) Copy content Toggle raw display
$23$ \( T + 100 \) Copy content Toggle raw display
$29$ \( T + 216 \) Copy content Toggle raw display
$31$ \( T - 208 \) Copy content Toggle raw display
$37$ \( T - 254 \) Copy content Toggle raw display
$41$ \( T - 206 \) Copy content Toggle raw display
$43$ \( T + 292 \) Copy content Toggle raw display
$47$ \( T + 320 \) Copy content Toggle raw display
$53$ \( T + 402 \) Copy content Toggle raw display
$59$ \( T - 370 \) Copy content Toggle raw display
$61$ \( T + 550 \) Copy content Toggle raw display
$67$ \( T + 728 \) Copy content Toggle raw display
$71$ \( T - 540 \) Copy content Toggle raw display
$73$ \( T + 604 \) Copy content Toggle raw display
$79$ \( T - 792 \) Copy content Toggle raw display
$83$ \( T - 404 \) Copy content Toggle raw display
$89$ \( T - 938 \) Copy content Toggle raw display
$97$ \( T + 56 \) Copy content Toggle raw display
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