Properties

Label 2166.2.a.j
Level $2166$
Weight $2$
Character orbit 2166.a
Self dual yes
Analytic conductor $17.296$
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] = [2166,2,Mod(1,2166)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2166, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2166.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.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: \(17.2955970778\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + ( - \beta + 3) q^{5} + q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + ( - \beta + 3) q^{5} + q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9} + (\beta - 3) q^{10} + ( - \beta + 2) q^{11} - q^{12} + ( - 4 \beta + 2) q^{13} + ( - \beta - 1) q^{14} + (\beta - 3) q^{15} + q^{16} + ( - 2 \beta + 4) q^{17} - q^{18} + ( - \beta + 3) q^{20} + ( - \beta - 1) q^{21} + (\beta - 2) q^{22} + (4 \beta - 4) q^{23} + q^{24} + ( - 5 \beta + 5) q^{25} + (4 \beta - 2) q^{26} - q^{27} + (\beta + 1) q^{28} + (\beta + 6) q^{29} + ( - \beta + 3) q^{30} + (4 \beta + 4) q^{31} - q^{32} + (\beta - 2) q^{33} + (2 \beta - 4) q^{34} + (\beta + 2) q^{35} + q^{36} + (8 \beta - 2) q^{37} + (4 \beta - 2) q^{39} + (\beta - 3) q^{40} + ( - 6 \beta + 8) q^{41} + (\beta + 1) q^{42} + ( - 4 \beta + 10) q^{43} + ( - \beta + 2) q^{44} + ( - \beta + 3) q^{45} + ( - 4 \beta + 4) q^{46} - 10 q^{47} - q^{48} + (3 \beta - 5) q^{49} + (5 \beta - 5) q^{50} + (2 \beta - 4) q^{51} + ( - 4 \beta + 2) q^{52} + ( - 7 \beta - 2) q^{53} + q^{54} + ( - 4 \beta + 7) q^{55} + ( - \beta - 1) q^{56} + ( - \beta - 6) q^{58} + (3 \beta - 10) q^{59} + (\beta - 3) q^{60} + (4 \beta - 2) q^{61} + ( - 4 \beta - 4) q^{62} + (\beta + 1) q^{63} + q^{64} + ( - 10 \beta + 10) q^{65} + ( - \beta + 2) q^{66} + (8 \beta - 4) q^{67} + ( - 2 \beta + 4) q^{68} + ( - 4 \beta + 4) q^{69} + ( - \beta - 2) q^{70} + ( - 8 \beta - 2) q^{71} - q^{72} + (7 \beta - 3) q^{73} + ( - 8 \beta + 2) q^{74} + (5 \beta - 5) q^{75} + q^{77} + ( - 4 \beta + 2) q^{78} + ( - \beta + 8) q^{79} + ( - \beta + 3) q^{80} + q^{81} + (6 \beta - 8) q^{82} + (5 \beta - 1) q^{83} + ( - \beta - 1) q^{84} + ( - 8 \beta + 14) q^{85} + (4 \beta - 10) q^{86} + ( - \beta - 6) q^{87} + (\beta - 2) q^{88} + (14 \beta - 4) q^{89} + (\beta - 3) q^{90} + ( - 6 \beta - 2) q^{91} + (4 \beta - 4) q^{92} + ( - 4 \beta - 4) q^{93} + 10 q^{94} + q^{96} + ( - 4 \beta + 2) q^{97} + ( - 3 \beta + 5) q^{98} + ( - \beta + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 5 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 5 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - 5 q^{10} + 3 q^{11} - 2 q^{12} - 3 q^{14} - 5 q^{15} + 2 q^{16} + 6 q^{17} - 2 q^{18} + 5 q^{20} - 3 q^{21} - 3 q^{22} - 4 q^{23} + 2 q^{24} + 5 q^{25} - 2 q^{27} + 3 q^{28} + 13 q^{29} + 5 q^{30} + 12 q^{31} - 2 q^{32} - 3 q^{33} - 6 q^{34} + 5 q^{35} + 2 q^{36} + 4 q^{37} - 5 q^{40} + 10 q^{41} + 3 q^{42} + 16 q^{43} + 3 q^{44} + 5 q^{45} + 4 q^{46} - 20 q^{47} - 2 q^{48} - 7 q^{49} - 5 q^{50} - 6 q^{51} - 11 q^{53} + 2 q^{54} + 10 q^{55} - 3 q^{56} - 13 q^{58} - 17 q^{59} - 5 q^{60} - 12 q^{62} + 3 q^{63} + 2 q^{64} + 10 q^{65} + 3 q^{66} + 6 q^{68} + 4 q^{69} - 5 q^{70} - 12 q^{71} - 2 q^{72} + q^{73} - 4 q^{74} - 5 q^{75} + 2 q^{77} + 15 q^{79} + 5 q^{80} + 2 q^{81} - 10 q^{82} + 3 q^{83} - 3 q^{84} + 20 q^{85} - 16 q^{86} - 13 q^{87} - 3 q^{88} + 6 q^{89} - 5 q^{90} - 10 q^{91} - 4 q^{92} - 12 q^{93} + 20 q^{94} + 2 q^{96} + 7 q^{98} + 3 q^{99}+O(q^{100}) \) 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 1.38197 1.00000 2.61803 −1.00000 1.00000 −1.38197
1.2 −1.00000 −1.00000 1.00000 3.61803 1.00000 0.381966 −1.00000 1.00000 −3.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(19\) \(-1\)

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 2166.2.a.j 2
3.b odd 2 1 6498.2.a.bf 2
19.b odd 2 1 2166.2.a.m yes 2
57.d even 2 1 6498.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2166.2.a.j 2 1.a even 1 1 trivial
2166.2.a.m yes 2 19.b odd 2 1
6498.2.a.z 2 57.d even 2 1
6498.2.a.bf 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{2} - 5T_{5} + 5 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{29}^{2} - 13T_{29} + 41 \) 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
$5$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 13T + 41 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$41$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$43$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 11T - 31 \) Copy content Toggle raw display
$59$ \( T^{2} + 17T + 61 \) Copy content Toggle raw display
$61$ \( T^{2} - 20 \) Copy content Toggle raw display
$67$ \( T^{2} - 80 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$73$ \( T^{2} - T - 61 \) Copy content Toggle raw display
$79$ \( T^{2} - 15T + 55 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 236 \) Copy content Toggle raw display
$97$ \( T^{2} - 20 \) Copy content Toggle raw display
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