Properties

Label 450.8.a.t
Level $450$
Weight $8$
Character orbit 450.a
Self dual yes
Analytic conductor $140.573$
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] = [450,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(140.573261468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
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)
 
\(f(q)\) \(=\) \( q + 8 q^{2} + 64 q^{4} - 104 q^{7} + 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + 64 q^{4} - 104 q^{7} + 512 q^{8} + 5148 q^{11} + 8602 q^{13} - 832 q^{14} + 4096 q^{16} + 20274 q^{17} + 45500 q^{19} + 41184 q^{22} - 72072 q^{23} + 68816 q^{26} - 6656 q^{28} - 231510 q^{29} - 80128 q^{31} + 32768 q^{32} + 162192 q^{34} - 104654 q^{37} + 364000 q^{38} - 584922 q^{41} + 795532 q^{43} + 329472 q^{44} - 576576 q^{46} + 425664 q^{47} - 812727 q^{49} + 550528 q^{52} + 1500798 q^{53} - 53248 q^{56} - 1852080 q^{58} - 246420 q^{59} + 893942 q^{61} - 641024 q^{62} + 262144 q^{64} + 2336836 q^{67} + 1297536 q^{68} + 203688 q^{71} + 3805702 q^{73} - 837232 q^{74} + 2912000 q^{76} - 535392 q^{77} + 5053040 q^{79} - 4679376 q^{82} - 45492 q^{83} + 6364256 q^{86} + 2635776 q^{88} - 980010 q^{89} - 894608 q^{91} - 4612608 q^{92} + 3405312 q^{94} + 5247646 q^{97} - 6501816 q^{98}+O(q^{100}) \) 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
8.00000 0 64.0000 0 0 −104.000 512.000 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 450.8.a.t 1
3.b odd 2 1 50.8.a.b 1
5.b even 2 1 90.8.a.a 1
5.c odd 4 2 450.8.c.p 2
12.b even 2 1 400.8.a.m 1
15.d odd 2 1 10.8.a.a 1
15.e even 4 2 50.8.b.d 2
60.h even 2 1 80.8.a.a 1
60.l odd 4 2 400.8.c.i 2
105.g even 2 1 490.8.a.b 1
120.i odd 2 1 320.8.a.c 1
120.m even 2 1 320.8.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.8.a.a 1 15.d odd 2 1
50.8.a.b 1 3.b odd 2 1
50.8.b.d 2 15.e even 4 2
80.8.a.a 1 60.h even 2 1
90.8.a.a 1 5.b even 2 1
320.8.a.c 1 120.i odd 2 1
320.8.a.f 1 120.m even 2 1
400.8.a.m 1 12.b even 2 1
400.8.c.i 2 60.l odd 4 2
450.8.a.t 1 1.a even 1 1 trivial
450.8.c.p 2 5.c odd 4 2
490.8.a.b 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 104 \) Copy content Toggle raw display
\( T_{11} - 5148 \) Copy content Toggle raw display
\( T_{17} - 20274 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 8 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 104 \) Copy content Toggle raw display
$11$ \( T - 5148 \) Copy content Toggle raw display
$13$ \( T - 8602 \) Copy content Toggle raw display
$17$ \( T - 20274 \) Copy content Toggle raw display
$19$ \( T - 45500 \) Copy content Toggle raw display
$23$ \( T + 72072 \) Copy content Toggle raw display
$29$ \( T + 231510 \) Copy content Toggle raw display
$31$ \( T + 80128 \) Copy content Toggle raw display
$37$ \( T + 104654 \) Copy content Toggle raw display
$41$ \( T + 584922 \) Copy content Toggle raw display
$43$ \( T - 795532 \) Copy content Toggle raw display
$47$ \( T - 425664 \) Copy content Toggle raw display
$53$ \( T - 1500798 \) Copy content Toggle raw display
$59$ \( T + 246420 \) Copy content Toggle raw display
$61$ \( T - 893942 \) Copy content Toggle raw display
$67$ \( T - 2336836 \) Copy content Toggle raw display
$71$ \( T - 203688 \) Copy content Toggle raw display
$73$ \( T - 3805702 \) Copy content Toggle raw display
$79$ \( T - 5053040 \) Copy content Toggle raw display
$83$ \( T + 45492 \) Copy content Toggle raw display
$89$ \( T + 980010 \) Copy content Toggle raw display
$97$ \( T - 5247646 \) Copy content Toggle raw display
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