Properties

Label 450.4.a.f
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 150)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + q^{7} - 8 q^{8} - 42 q^{11} + 67 q^{13} - 2 q^{14} + 16 q^{16} + 54 q^{17} - 115 q^{19} + 84 q^{22} - 162 q^{23} - 134 q^{26} + 4 q^{28} + 210 q^{29} - 193 q^{31} - 32 q^{32} - 108 q^{34} + 286 q^{37} + 230 q^{38} - 12 q^{41} - 263 q^{43} - 168 q^{44} + 324 q^{46} + 414 q^{47} - 342 q^{49} + 268 q^{52} - 192 q^{53} - 8 q^{56} - 420 q^{58} - 690 q^{59} - 733 q^{61} + 386 q^{62} + 64 q^{64} - 299 q^{67} + 216 q^{68} + 228 q^{71} - 938 q^{73} - 572 q^{74} - 460 q^{76} - 42 q^{77} - 160 q^{79} + 24 q^{82} - 462 q^{83} + 526 q^{86} + 336 q^{88} + 240 q^{89} + 67 q^{91} - 648 q^{92} - 828 q^{94} + 511 q^{97} + 684 q^{98}+O(q^{100}) \) 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 1.00000 −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.f 1
3.b odd 2 1 150.4.a.h yes 1
5.b even 2 1 450.4.a.o 1
5.c odd 4 2 450.4.c.a 2
12.b even 2 1 1200.4.a.i 1
15.d odd 2 1 150.4.a.a 1
15.e even 4 2 150.4.c.e 2
60.h even 2 1 1200.4.a.bb 1
60.l odd 4 2 1200.4.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.a.a 1 15.d odd 2 1
150.4.a.h yes 1 3.b odd 2 1
150.4.c.e 2 15.e even 4 2
450.4.a.f 1 1.a even 1 1 trivial
450.4.a.o 1 5.b even 2 1
450.4.c.a 2 5.c odd 4 2
1200.4.a.i 1 12.b even 2 1
1200.4.a.bb 1 60.h even 2 1
1200.4.f.c 2 60.l odd 4 2

Hecke kernels

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

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} + 42 \) Copy content Toggle raw display
\( T_{17} - 54 \) 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
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 42 \) Copy content Toggle raw display
$13$ \( T - 67 \) Copy content Toggle raw display
$17$ \( T - 54 \) Copy content Toggle raw display
$19$ \( T + 115 \) Copy content Toggle raw display
$23$ \( T + 162 \) Copy content Toggle raw display
$29$ \( T - 210 \) Copy content Toggle raw display
$31$ \( T + 193 \) Copy content Toggle raw display
$37$ \( T - 286 \) Copy content Toggle raw display
$41$ \( T + 12 \) Copy content Toggle raw display
$43$ \( T + 263 \) Copy content Toggle raw display
$47$ \( T - 414 \) Copy content Toggle raw display
$53$ \( T + 192 \) Copy content Toggle raw display
$59$ \( T + 690 \) Copy content Toggle raw display
$61$ \( T + 733 \) Copy content Toggle raw display
$67$ \( T + 299 \) Copy content Toggle raw display
$71$ \( T - 228 \) Copy content Toggle raw display
$73$ \( T + 938 \) Copy content Toggle raw display
$79$ \( T + 160 \) Copy content Toggle raw display
$83$ \( T + 462 \) Copy content Toggle raw display
$89$ \( T - 240 \) Copy content Toggle raw display
$97$ \( T - 511 \) Copy content Toggle raw display
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