Properties

Label 5550.2.a.bt
Level $5550$
Weight $2$
Character orbit 5550.a
Self dual yes
Analytic conductor $44.317$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} + (\beta - 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + q^{6} + (\beta - 1) q^{7} - q^{8} + q^{9} + ( - \beta - 1) q^{11} - q^{12} + ( - \beta - 3) q^{13} + ( - \beta + 1) q^{14} + q^{16} + (\beta - 1) q^{17} - q^{18} + 3 q^{19} + ( - \beta + 1) q^{21} + (\beta + 1) q^{22} - q^{23} + q^{24} + (\beta + 3) q^{26} - q^{27} + (\beta - 1) q^{28} + (\beta + 6) q^{29} - 2 q^{31} - q^{32} + (\beta + 1) q^{33} + ( - \beta + 1) q^{34} + q^{36} + q^{37} - 3 q^{38} + (\beta + 3) q^{39} + (\beta + 2) q^{41} + (\beta - 1) q^{42} + ( - 2 \beta + 2) q^{43} + ( - \beta - 1) q^{44} + q^{46} + ( - \beta + 4) q^{47} - q^{48} - 2 \beta q^{49} + ( - \beta + 1) q^{51} + ( - \beta - 3) q^{52} + 3 q^{53} + q^{54} + ( - \beta + 1) q^{56} - 3 q^{57} + ( - \beta - 6) q^{58} + (3 \beta + 4) q^{59} + (2 \beta - 10) q^{61} + 2 q^{62} + (\beta - 1) q^{63} + q^{64} + ( - \beta - 1) q^{66} + (2 \beta - 2) q^{67} + (\beta - 1) q^{68} + q^{69} + (3 \beta + 6) q^{71} - q^{72} + ( - 4 \beta - 1) q^{73} - q^{74} + 3 q^{76} - 5 q^{77} + ( - \beta - 3) q^{78} + ( - 3 \beta + 6) q^{79} + q^{81} + ( - \beta - 2) q^{82} + ( - \beta - 1) q^{83} + ( - \beta + 1) q^{84} + (2 \beta - 2) q^{86} + ( - \beta - 6) q^{87} + (\beta + 1) q^{88} + ( - 3 \beta - 7) q^{89} + ( - 2 \beta - 3) q^{91} - q^{92} + 2 q^{93} + (\beta - 4) q^{94} + q^{96} + (3 \beta + 4) q^{97} + 2 \beta q^{98} + ( - \beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{11} - 2 q^{12} - 6 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 6 q^{19} + 2 q^{21} + 2 q^{22} - 2 q^{23} + 2 q^{24} + 6 q^{26} - 2 q^{27} - 2 q^{28} + 12 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 2 q^{37} - 6 q^{38} + 6 q^{39} + 4 q^{41} - 2 q^{42} + 4 q^{43} - 2 q^{44} + 2 q^{46} + 8 q^{47} - 2 q^{48} + 2 q^{51} - 6 q^{52} + 6 q^{53} + 2 q^{54} + 2 q^{56} - 6 q^{57} - 12 q^{58} + 8 q^{59} - 20 q^{61} + 4 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{66} - 4 q^{67} - 2 q^{68} + 2 q^{69} + 12 q^{71} - 2 q^{72} - 2 q^{73} - 2 q^{74} + 6 q^{76} - 10 q^{77} - 6 q^{78} + 12 q^{79} + 2 q^{81} - 4 q^{82} - 2 q^{83} + 2 q^{84} - 4 q^{86} - 12 q^{87} + 2 q^{88} - 14 q^{89} - 6 q^{91} - 2 q^{92} + 4 q^{93} - 8 q^{94} + 2 q^{96} + 8 q^{97} - 2 q^{99}+O(q^{100}) \) 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
−2.44949
2.44949
−1.00000 −1.00000 1.00000 0 1.00000 −3.44949 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 1.44949 −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.bt 2
5.b even 2 1 5550.2.a.cc 2
5.c odd 4 2 1110.2.d.h 4
15.e even 4 2 3330.2.d.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.h 4 5.c odd 4 2
3330.2.d.h 4 15.e even 4 2
5550.2.a.bt 2 1.a even 1 1 trivial
5550.2.a.cc 2 5.b even 2 1

Hecke kernels

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

\( T_{7}^{2} + 2T_{7} - 5 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 5 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 3 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} - 5 \) 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
$7$ \( T^{2} + 2T - 5 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 5 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 3 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 5 \) Copy content Toggle raw display
$19$ \( (T - 3)^{2} \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 12T + 30 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 10 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 38 \) Copy content Toggle raw display
$61$ \( T^{2} + 20T + 76 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 20 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T - 18 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T - 95 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T - 18 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 5 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T - 5 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 38 \) Copy content Toggle raw display
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