Properties

Label 8550.2.a.bv
Level $8550$
Weight $2$
Character orbit 8550.a
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8550,2,Mod(1,8550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8550, 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("8550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.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: \(68.2720937282\)
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 2850)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta - 2) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (\beta - 2) q^{7} + q^{8} + (\beta - 1) q^{11} - \beta q^{13} + (\beta - 2) q^{14} + q^{16} + ( - \beta + 2) q^{17} + q^{19} + (\beta - 1) q^{22} - q^{23} - \beta q^{26} + (\beta - 2) q^{28} + ( - 3 \beta - 3) q^{29} - 3 q^{31} + q^{32} + ( - \beta + 2) q^{34} + ( - 4 \beta - 2) q^{37} + q^{38} + ( - 2 \beta - 4) q^{41} - \beta q^{43} + (\beta - 1) q^{44} - q^{46} + (4 \beta + 2) q^{47} + ( - 4 \beta + 3) q^{49} - \beta q^{52} + (\beta - 5) q^{53} + (\beta - 2) q^{56} + ( - 3 \beta - 3) q^{58} + (\beta - 4) q^{59} + (\beta - 7) q^{61} - 3 q^{62} + q^{64} + (5 \beta - 3) q^{67} + ( - \beta + 2) q^{68} + (\beta + 4) q^{71} + q^{73} + ( - 4 \beta - 2) q^{74} + q^{76} + ( - 3 \beta + 8) q^{77} - 5 q^{79} + ( - 2 \beta - 4) q^{82} + ( - 3 \beta + 1) q^{83} - \beta q^{86} + (\beta - 1) q^{88} + ( - 2 \beta - 7) q^{89} + (2 \beta - 6) q^{91} - q^{92} + (4 \beta + 2) q^{94} + ( - \beta - 4) q^{97} + ( - 4 \beta + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 2 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{17} + 2 q^{19} - 2 q^{22} - 2 q^{23} - 4 q^{28} - 6 q^{29} - 6 q^{31} + 2 q^{32} + 4 q^{34} - 4 q^{37} + 2 q^{38} - 8 q^{41} - 2 q^{44} - 2 q^{46} + 4 q^{47} + 6 q^{49} - 10 q^{53} - 4 q^{56} - 6 q^{58} - 8 q^{59} - 14 q^{61} - 6 q^{62} + 2 q^{64} - 6 q^{67} + 4 q^{68} + 8 q^{71} + 2 q^{73} - 4 q^{74} + 2 q^{76} + 16 q^{77} - 10 q^{79} - 8 q^{82} + 2 q^{83} - 2 q^{88} - 14 q^{89} - 12 q^{91} - 2 q^{92} + 4 q^{94} - 8 q^{97} + 6 q^{98}+O(q^{100}) \) 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 0 1.00000 0 0 −4.44949 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 0.449490 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(19\) \(-1\)

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 8550.2.a.bv 2
3.b odd 2 1 2850.2.a.bc 2
5.b even 2 1 8550.2.a.bu 2
15.d odd 2 1 2850.2.a.bj yes 2
15.e even 4 2 2850.2.d.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bc 2 3.b odd 2 1
2850.2.a.bj yes 2 15.d odd 2 1
2850.2.d.w 4 15.e even 4 2
8550.2.a.bu 2 5.b even 2 1
8550.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{2} + 4T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 5 \) Copy content Toggle raw display
\( T_{13}^{2} - 6 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 2 \) Copy content Toggle raw display
\( T_{23} + 1 \) Copy content Toggle raw display
\( T_{53}^{2} + 10T_{53} + 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 5 \) Copy content Toggle raw display
$13$ \( T^{2} - 6 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 45 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 92 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} - 6 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 92 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 19 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 43 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T - 141 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T + 10 \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( (T + 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 53 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 25 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
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