Properties

Label 550.6.a.f
Level $550$
Weight $6$
Character orbit 550.a
Self dual yes
Analytic conductor $88.211$
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] = [550,6,Mod(1,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 550.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: \(88.2111008971\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - q^{3} + 16 q^{4} - 4 q^{6} + 166 q^{7} + 64 q^{8} - 242 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - q^{3} + 16 q^{4} - 4 q^{6} + 166 q^{7} + 64 q^{8} - 242 q^{9} - 121 q^{11} - 16 q^{12} - 692 q^{13} + 664 q^{14} + 256 q^{16} + 738 q^{17} - 968 q^{18} + 1424 q^{19} - 166 q^{21} - 484 q^{22} + 1779 q^{23} - 64 q^{24} - 2768 q^{26} + 485 q^{27} + 2656 q^{28} - 2064 q^{29} + 6245 q^{31} + 1024 q^{32} + 121 q^{33} + 2952 q^{34} - 3872 q^{36} + 14785 q^{37} + 5696 q^{38} + 692 q^{39} + 5304 q^{41} - 664 q^{42} - 17798 q^{43} - 1936 q^{44} + 7116 q^{46} + 17184 q^{47} - 256 q^{48} + 10749 q^{49} - 738 q^{51} - 11072 q^{52} + 30726 q^{53} + 1940 q^{54} + 10624 q^{56} - 1424 q^{57} - 8256 q^{58} - 34989 q^{59} - 45940 q^{61} + 24980 q^{62} - 40172 q^{63} + 4096 q^{64} + 484 q^{66} - 25343 q^{67} + 11808 q^{68} - 1779 q^{69} + 13311 q^{71} - 15488 q^{72} + 53260 q^{73} + 59140 q^{74} + 22784 q^{76} - 20086 q^{77} + 2768 q^{78} + 77234 q^{79} + 58321 q^{81} + 21216 q^{82} - 55014 q^{83} - 2656 q^{84} - 71192 q^{86} + 2064 q^{87} - 7744 q^{88} + 125415 q^{89} - 114872 q^{91} + 28464 q^{92} - 6245 q^{93} + 68736 q^{94} - 1024 q^{96} + 88807 q^{97} + 42996 q^{98} + 29282 q^{99}+O(q^{100}) \) 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
4.00000 −1.00000 16.0000 0 −4.00000 166.000 64.0000 −242.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 550.6.a.f 1
5.b even 2 1 22.6.a.b 1
5.c odd 4 2 550.6.b.f 2
15.d odd 2 1 198.6.a.i 1
20.d odd 2 1 176.6.a.b 1
35.c odd 2 1 1078.6.a.a 1
40.e odd 2 1 704.6.a.f 1
40.f even 2 1 704.6.a.e 1
55.d odd 2 1 242.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 5.b even 2 1
176.6.a.b 1 20.d odd 2 1
198.6.a.i 1 15.d odd 2 1
242.6.a.d 1 55.d odd 2 1
550.6.a.f 1 1.a even 1 1 trivial
550.6.b.f 2 5.c odd 4 2
704.6.a.e 1 40.f even 2 1
704.6.a.f 1 40.e odd 2 1
1078.6.a.a 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(550))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 166 \) Copy content Toggle raw display
$11$ \( T + 121 \) Copy content Toggle raw display
$13$ \( T + 692 \) Copy content Toggle raw display
$17$ \( T - 738 \) Copy content Toggle raw display
$19$ \( T - 1424 \) Copy content Toggle raw display
$23$ \( T - 1779 \) Copy content Toggle raw display
$29$ \( T + 2064 \) Copy content Toggle raw display
$31$ \( T - 6245 \) Copy content Toggle raw display
$37$ \( T - 14785 \) Copy content Toggle raw display
$41$ \( T - 5304 \) Copy content Toggle raw display
$43$ \( T + 17798 \) Copy content Toggle raw display
$47$ \( T - 17184 \) Copy content Toggle raw display
$53$ \( T - 30726 \) Copy content Toggle raw display
$59$ \( T + 34989 \) Copy content Toggle raw display
$61$ \( T + 45940 \) Copy content Toggle raw display
$67$ \( T + 25343 \) Copy content Toggle raw display
$71$ \( T - 13311 \) Copy content Toggle raw display
$73$ \( T - 53260 \) Copy content Toggle raw display
$79$ \( T - 77234 \) Copy content Toggle raw display
$83$ \( T + 55014 \) Copy content Toggle raw display
$89$ \( T - 125415 \) Copy content Toggle raw display
$97$ \( T - 88807 \) Copy content Toggle raw display
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