Properties

Label 450.8.a.c
Level $450$
Weight $8$
Character orbit 450.a
Self dual yes
Analytic conductor $140.573$
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] = [450,8,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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} + 64 q^{4} - 1016 q^{7} - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 64 q^{4} - 1016 q^{7} - 512 q^{8} - 1092 q^{11} - 1382 q^{13} + 8128 q^{14} + 4096 q^{16} + 14706 q^{17} - 39940 q^{19} + 8736 q^{22} + 68712 q^{23} + 11056 q^{26} - 65024 q^{28} + 102570 q^{29} + 227552 q^{31} - 32768 q^{32} - 117648 q^{34} - 160526 q^{37} + 319520 q^{38} - 10842 q^{41} + 630748 q^{43} - 69888 q^{44} - 549696 q^{46} + 472656 q^{47} + 208713 q^{49} - 88448 q^{52} - 1494018 q^{53} + 520192 q^{56} - 820560 q^{58} - 2640660 q^{59} + 827702 q^{61} - 1820416 q^{62} + 262144 q^{64} + 126004 q^{67} + 941184 q^{68} + 1414728 q^{71} - 980282 q^{73} + 1284208 q^{74} - 2556160 q^{76} + 1109472 q^{77} - 3566800 q^{79} + 86736 q^{82} + 5672892 q^{83} - 5045984 q^{86} + 559104 q^{88} + 11951190 q^{89} + 1404112 q^{91} + 4397568 q^{92} - 3781248 q^{94} - 8682146 q^{97} - 1669704 q^{98}+O(q^{100}) \) 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 −1016.00 −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.c 1
3.b odd 2 1 50.8.a.g 1
5.b even 2 1 18.8.a.b 1
5.c odd 4 2 450.8.c.g 2
12.b even 2 1 400.8.a.l 1
15.d odd 2 1 2.8.a.a 1
15.e even 4 2 50.8.b.c 2
20.d odd 2 1 144.8.a.i 1
40.e odd 2 1 576.8.a.f 1
40.f even 2 1 576.8.a.g 1
45.h odd 6 2 162.8.c.l 2
45.j even 6 2 162.8.c.a 2
60.h even 2 1 16.8.a.b 1
60.l odd 4 2 400.8.c.j 2
105.g even 2 1 98.8.a.a 1
105.o odd 6 2 98.8.c.d 2
105.p even 6 2 98.8.c.e 2
120.i odd 2 1 64.8.a.c 1
120.m even 2 1 64.8.a.e 1
165.d even 2 1 242.8.a.e 1
195.e odd 2 1 338.8.a.d 1
195.n even 4 2 338.8.b.d 2
240.t even 4 2 256.8.b.f 2
240.bm odd 4 2 256.8.b.b 2
255.h odd 2 1 578.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a 1 15.d odd 2 1
16.8.a.b 1 60.h even 2 1
18.8.a.b 1 5.b even 2 1
50.8.a.g 1 3.b odd 2 1
50.8.b.c 2 15.e even 4 2
64.8.a.c 1 120.i odd 2 1
64.8.a.e 1 120.m even 2 1
98.8.a.a 1 105.g even 2 1
98.8.c.d 2 105.o odd 6 2
98.8.c.e 2 105.p even 6 2
144.8.a.i 1 20.d odd 2 1
162.8.c.a 2 45.j even 6 2
162.8.c.l 2 45.h odd 6 2
242.8.a.e 1 165.d even 2 1
256.8.b.b 2 240.bm odd 4 2
256.8.b.f 2 240.t even 4 2
338.8.a.d 1 195.e odd 2 1
338.8.b.d 2 195.n even 4 2
400.8.a.l 1 12.b even 2 1
400.8.c.j 2 60.l odd 4 2
450.8.a.c 1 1.a even 1 1 trivial
450.8.c.g 2 5.c odd 4 2
576.8.a.f 1 40.e odd 2 1
576.8.a.g 1 40.f even 2 1
578.8.a.b 1 255.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 1016 \) Copy content Toggle raw display
\( T_{11} + 1092 \) Copy content Toggle raw display
\( T_{17} - 14706 \) 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
$7$ \( T + 1016 \) Copy content Toggle raw display
$11$ \( T + 1092 \) Copy content Toggle raw display
$13$ \( T + 1382 \) Copy content Toggle raw display
$17$ \( T - 14706 \) Copy content Toggle raw display
$19$ \( T + 39940 \) Copy content Toggle raw display
$23$ \( T - 68712 \) Copy content Toggle raw display
$29$ \( T - 102570 \) Copy content Toggle raw display
$31$ \( T - 227552 \) Copy content Toggle raw display
$37$ \( T + 160526 \) Copy content Toggle raw display
$41$ \( T + 10842 \) Copy content Toggle raw display
$43$ \( T - 630748 \) Copy content Toggle raw display
$47$ \( T - 472656 \) Copy content Toggle raw display
$53$ \( T + 1494018 \) Copy content Toggle raw display
$59$ \( T + 2640660 \) Copy content Toggle raw display
$61$ \( T - 827702 \) Copy content Toggle raw display
$67$ \( T - 126004 \) Copy content Toggle raw display
$71$ \( T - 1414728 \) Copy content Toggle raw display
$73$ \( T + 980282 \) Copy content Toggle raw display
$79$ \( T + 3566800 \) Copy content Toggle raw display
$83$ \( T - 5672892 \) Copy content Toggle raw display
$89$ \( T - 11951190 \) Copy content Toggle raw display
$97$ \( T + 8682146 \) Copy content Toggle raw display
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