Properties

Label 450.4.a.b
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - 32 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} - 32 q^{7} - 8 q^{8} + 60 q^{11} + 34 q^{13} + 64 q^{14} + 16 q^{16} + 42 q^{17} - 76 q^{19} - 120 q^{22} - 68 q^{26} - 128 q^{28} - 6 q^{29} - 232 q^{31} - 32 q^{32} - 84 q^{34} - 134 q^{37} + 152 q^{38} - 234 q^{41} + 412 q^{43} + 240 q^{44} - 360 q^{47} + 681 q^{49} + 136 q^{52} + 222 q^{53} + 256 q^{56} + 12 q^{58} - 660 q^{59} - 490 q^{61} + 464 q^{62} + 64 q^{64} - 812 q^{67} + 168 q^{68} - 120 q^{71} - 746 q^{73} + 268 q^{74} - 304 q^{76} - 1920 q^{77} + 152 q^{79} + 468 q^{82} - 804 q^{83} - 824 q^{86} - 480 q^{88} + 678 q^{89} - 1088 q^{91} + 720 q^{94} - 194 q^{97} - 1362 q^{98}+O(q^{100}) \) 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 −32.0000 −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.b 1
3.b odd 2 1 150.4.a.e 1
5.b even 2 1 90.4.a.d 1
5.c odd 4 2 450.4.c.k 2
12.b even 2 1 1200.4.a.bk 1
15.d odd 2 1 30.4.a.a 1
15.e even 4 2 150.4.c.a 2
20.d odd 2 1 720.4.a.b 1
45.h odd 6 2 810.4.e.m 2
45.j even 6 2 810.4.e.e 2
60.h even 2 1 240.4.a.c 1
60.l odd 4 2 1200.4.f.u 2
105.g even 2 1 1470.4.a.a 1
120.i odd 2 1 960.4.a.j 1
120.m even 2 1 960.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.a 1 15.d odd 2 1
90.4.a.d 1 5.b even 2 1
150.4.a.e 1 3.b odd 2 1
150.4.c.a 2 15.e even 4 2
240.4.a.c 1 60.h even 2 1
450.4.a.b 1 1.a even 1 1 trivial
450.4.c.k 2 5.c odd 4 2
720.4.a.b 1 20.d odd 2 1
810.4.e.e 2 45.j even 6 2
810.4.e.m 2 45.h odd 6 2
960.4.a.j 1 120.i odd 2 1
960.4.a.s 1 120.m even 2 1
1200.4.a.bk 1 12.b even 2 1
1200.4.f.u 2 60.l odd 4 2
1470.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 32 \) Copy content Toggle raw display
\( T_{11} - 60 \) Copy content Toggle raw display
\( T_{17} - 42 \) 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
$7$ \( T + 32 \) Copy content Toggle raw display
$11$ \( T - 60 \) Copy content Toggle raw display
$13$ \( T - 34 \) Copy content Toggle raw display
$17$ \( T - 42 \) Copy content Toggle raw display
$19$ \( T + 76 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 232 \) Copy content Toggle raw display
$37$ \( T + 134 \) Copy content Toggle raw display
$41$ \( T + 234 \) Copy content Toggle raw display
$43$ \( T - 412 \) Copy content Toggle raw display
$47$ \( T + 360 \) Copy content Toggle raw display
$53$ \( T - 222 \) Copy content Toggle raw display
$59$ \( T + 660 \) Copy content Toggle raw display
$61$ \( T + 490 \) Copy content Toggle raw display
$67$ \( T + 812 \) Copy content Toggle raw display
$71$ \( T + 120 \) Copy content Toggle raw display
$73$ \( T + 746 \) Copy content Toggle raw display
$79$ \( T - 152 \) Copy content Toggle raw display
$83$ \( T + 804 \) Copy content Toggle raw display
$89$ \( T - 678 \) Copy content Toggle raw display
$97$ \( T + 194 \) Copy content Toggle raw display
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