Properties

Label 5550.2.a.bz
Level $5550$
Weight $2$
Character orbit 5550.a
Self dual yes
Analytic conductor $44.317$
Analytic rank $0$
Dimension $2$
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] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.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: \(44.3169731218\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1110)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + 2 q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{6} + 2 q^{7} + q^{8} + q^{9} + ( - \beta + 3) q^{11} - q^{12} + 2 \beta q^{13} + 2 q^{14} + q^{16} - 2 \beta q^{17} + q^{18} + (\beta - 5) q^{19} - 2 q^{21} + ( - \beta + 3) q^{22} + 4 q^{23} - q^{24} + 2 \beta q^{26} - q^{27} + 2 q^{28} + 4 q^{29} + (2 \beta + 2) q^{31} + q^{32} + (\beta - 3) q^{33} - 2 \beta q^{34} + q^{36} + q^{37} + (\beta - 5) q^{38} - 2 \beta q^{39} + 2 \beta q^{41} - 2 q^{42} + ( - 2 \beta + 6) q^{43} + ( - \beta + 3) q^{44} + 4 q^{46} + ( - 3 \beta + 3) q^{47} - q^{48} - 3 q^{49} + 2 \beta q^{51} + 2 \beta q^{52} + ( - 2 \beta + 8) q^{53} - q^{54} + 2 q^{56} + ( - \beta + 5) q^{57} + 4 q^{58} + ( - 4 \beta - 2) q^{59} + ( - \beta + 3) q^{61} + (2 \beta + 2) q^{62} + 2 q^{63} + q^{64} + (\beta - 3) q^{66} + (2 \beta - 6) q^{67} - 2 \beta q^{68} - 4 q^{69} + (2 \beta + 2) q^{71} + q^{72} + ( - 2 \beta + 6) q^{73} + q^{74} + (\beta - 5) q^{76} + ( - 2 \beta + 6) q^{77} - 2 \beta q^{78} + (2 \beta + 2) q^{79} + q^{81} + 2 \beta q^{82} + (6 \beta - 2) q^{83} - 2 q^{84} + ( - 2 \beta + 6) q^{86} - 4 q^{87} + ( - \beta + 3) q^{88} + ( - 2 \beta - 12) q^{89} + 4 \beta q^{91} + 4 q^{92} + ( - 2 \beta - 2) q^{93} + ( - 3 \beta + 3) q^{94} - q^{96} + (5 \beta + 3) q^{97} - 3 q^{98} + ( - \beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 4 q^{14} + 2 q^{16} + 2 q^{18} - 10 q^{19} - 4 q^{21} + 6 q^{22} + 8 q^{23} - 2 q^{24} - 2 q^{27} + 4 q^{28} + 8 q^{29} + 4 q^{31} + 2 q^{32} - 6 q^{33} + 2 q^{36} + 2 q^{37} - 10 q^{38} - 4 q^{42} + 12 q^{43} + 6 q^{44} + 8 q^{46} + 6 q^{47} - 2 q^{48} - 6 q^{49} + 16 q^{53} - 2 q^{54} + 4 q^{56} + 10 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{61} + 4 q^{62} + 4 q^{63} + 2 q^{64} - 6 q^{66} - 12 q^{67} - 8 q^{69} + 4 q^{71} + 2 q^{72} + 12 q^{73} + 2 q^{74} - 10 q^{76} + 12 q^{77} + 4 q^{79} + 2 q^{81} - 4 q^{83} - 4 q^{84} + 12 q^{86} - 8 q^{87} + 6 q^{88} - 24 q^{89} + 8 q^{92} - 4 q^{93} + 6 q^{94} - 2 q^{96} + 6 q^{97} - 6 q^{98} + 6 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
1.00000 −1.00000 1.00000 0 −1.00000 2.00000 1.00000 1.00000 0
1.2 1.00000 −1.00000 1.00000 0 −1.00000 2.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(37\) \(-1\)

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 5550.2.a.bz 2
5.b even 2 1 5550.2.a.bu 2
5.c odd 4 2 1110.2.d.f 4
15.e even 4 2 3330.2.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.f 4 5.c odd 4 2
3330.2.d.j 4 15.e even 4 2
5550.2.a.bu 2 5.b even 2 1
5550.2.a.bz 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5550))\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 20 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 176 \) Copy content Toggle raw display
$89$ \( T^{2} + 24T + 124 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 116 \) Copy content Toggle raw display
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