Properties

Label 550.4.a.k
Level $550$
Weight $4$
Character orbit 550.a
Self dual yes
Analytic conductor $32.451$
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] = [550,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(32.4510505032\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} + 8 q^{7} + 8 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} + 8 q^{7} + 8 q^{8} - 11 q^{9} - 11 q^{11} - 16 q^{12} + 50 q^{13} + 16 q^{14} + 16 q^{16} - 130 q^{17} - 22 q^{18} - 108 q^{19} - 32 q^{21} - 22 q^{22} + 96 q^{23} - 32 q^{24} + 100 q^{26} + 152 q^{27} + 32 q^{28} + 142 q^{29} + 40 q^{31} + 32 q^{32} + 44 q^{33} - 260 q^{34} - 44 q^{36} - 382 q^{37} - 216 q^{38} - 200 q^{39} - 118 q^{41} - 64 q^{42} - 220 q^{43} - 44 q^{44} + 192 q^{46} - 520 q^{47} - 64 q^{48} - 279 q^{49} + 520 q^{51} + 200 q^{52} - 238 q^{53} + 304 q^{54} + 64 q^{56} + 432 q^{57} + 284 q^{58} - 852 q^{59} + 190 q^{61} + 80 q^{62} - 88 q^{63} + 64 q^{64} + 88 q^{66} + 12 q^{67} - 520 q^{68} - 384 q^{69} - 112 q^{71} - 88 q^{72} + 6 q^{73} - 764 q^{74} - 432 q^{76} - 88 q^{77} - 400 q^{78} + 304 q^{79} - 311 q^{81} - 236 q^{82} - 820 q^{83} - 128 q^{84} - 440 q^{86} - 568 q^{87} - 88 q^{88} + 202 q^{89} + 400 q^{91} + 384 q^{92} - 160 q^{93} - 1040 q^{94} - 128 q^{96} + 1406 q^{97} - 558 q^{98} + 121 q^{99}+O(q^{100}) \) 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
2.00000 −4.00000 4.00000 0 −8.00000 8.00000 8.00000 −11.0000 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 550.4.a.k 1
5.b even 2 1 22.4.a.b 1
5.c odd 4 2 550.4.b.b 2
15.d odd 2 1 198.4.a.d 1
20.d odd 2 1 176.4.a.b 1
35.c odd 2 1 1078.4.a.a 1
40.e odd 2 1 704.4.a.i 1
40.f even 2 1 704.4.a.d 1
55.d odd 2 1 242.4.a.f 1
55.h odd 10 4 242.4.c.b 4
55.j even 10 4 242.4.c.h 4
60.h even 2 1 1584.4.a.b 1
165.d even 2 1 2178.4.a.a 1
220.g even 2 1 1936.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.a.b 1 5.b even 2 1
176.4.a.b 1 20.d odd 2 1
198.4.a.d 1 15.d odd 2 1
242.4.a.f 1 55.d odd 2 1
242.4.c.b 4 55.h odd 10 4
242.4.c.h 4 55.j even 10 4
550.4.a.k 1 1.a even 1 1 trivial
550.4.b.b 2 5.c odd 4 2
704.4.a.d 1 40.f even 2 1
704.4.a.i 1 40.e odd 2 1
1078.4.a.a 1 35.c odd 2 1
1584.4.a.b 1 60.h even 2 1
1936.4.a.g 1 220.g even 2 1
2178.4.a.a 1 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(550))\):

\( T_{3} + 4 \) Copy content Toggle raw display
\( T_{7} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 4 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 8 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 50 \) Copy content Toggle raw display
$17$ \( T + 130 \) Copy content Toggle raw display
$19$ \( T + 108 \) Copy content Toggle raw display
$23$ \( T - 96 \) Copy content Toggle raw display
$29$ \( T - 142 \) Copy content Toggle raw display
$31$ \( T - 40 \) Copy content Toggle raw display
$37$ \( T + 382 \) Copy content Toggle raw display
$41$ \( T + 118 \) Copy content Toggle raw display
$43$ \( T + 220 \) Copy content Toggle raw display
$47$ \( T + 520 \) Copy content Toggle raw display
$53$ \( T + 238 \) Copy content Toggle raw display
$59$ \( T + 852 \) Copy content Toggle raw display
$61$ \( T - 190 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 112 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T - 304 \) Copy content Toggle raw display
$83$ \( T + 820 \) Copy content Toggle raw display
$89$ \( T - 202 \) Copy content Toggle raw display
$97$ \( T - 1406 \) Copy content Toggle raw display
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