Properties

Label 5610.2.a.bo
Level $5610$
Weight $2$
Character orbit 5610.a
Self dual yes
Analytic conductor $44.796$
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] = [5610,2,Mod(1,5610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.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: \(44.7960755339\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - \beta q^{7} - q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - \beta q^{7} - q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + ( - \beta + 2) q^{13} + \beta q^{14} + q^{15} + q^{16} + q^{17} - q^{18} + (\beta - 2) q^{19} - q^{20} + \beta q^{21} - q^{22} + (\beta - 6) q^{23} + q^{24} + q^{25} + (\beta - 2) q^{26} - q^{27} - \beta q^{28} + 2 q^{29} - q^{30} + ( - \beta - 2) q^{31} - q^{32} - q^{33} - q^{34} + \beta q^{35} + q^{36} + ( - \beta + 8) q^{37} + ( - \beta + 2) q^{38} + (\beta - 2) q^{39} + q^{40} + (2 \beta - 4) q^{41} - \beta q^{42} + (2 \beta - 2) q^{43} + q^{44} - q^{45} + ( - \beta + 6) q^{46} + (2 \beta - 4) q^{47} - q^{48} + (\beta + 3) q^{49} - q^{50} - q^{51} + ( - \beta + 2) q^{52} + 2 q^{53} + q^{54} - q^{55} + \beta q^{56} + ( - \beta + 2) q^{57} - 2 q^{58} + ( - 2 \beta - 6) q^{59} + q^{60} + (\beta + 4) q^{61} + (\beta + 2) q^{62} - \beta q^{63} + q^{64} + (\beta - 2) q^{65} + q^{66} + (\beta - 12) q^{67} + q^{68} + ( - \beta + 6) q^{69} - \beta q^{70} + (4 \beta - 2) q^{71} - q^{72} - 8 q^{73} + (\beta - 8) q^{74} - q^{75} + (\beta - 2) q^{76} - \beta q^{77} + ( - \beta + 2) q^{78} + 2 \beta q^{79} - q^{80} + q^{81} + ( - 2 \beta + 4) q^{82} + ( - \beta - 6) q^{83} + \beta q^{84} - q^{85} + ( - 2 \beta + 2) q^{86} - 2 q^{87} - q^{88} + 14 q^{89} + q^{90} + ( - \beta + 10) q^{91} + (\beta - 6) q^{92} + (\beta + 2) q^{93} + ( - 2 \beta + 4) q^{94} + ( - \beta + 2) q^{95} + q^{96} + (\beta + 6) q^{97} + ( - \beta - 3) q^{98} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} + 3 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 3 q^{19} - 2 q^{20} + q^{21} - 2 q^{22} - 11 q^{23} + 2 q^{24} + 2 q^{25} - 3 q^{26} - 2 q^{27} - q^{28} + 4 q^{29} - 2 q^{30} - 5 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} + q^{35} + 2 q^{36} + 15 q^{37} + 3 q^{38} - 3 q^{39} + 2 q^{40} - 6 q^{41} - q^{42} - 2 q^{43} + 2 q^{44} - 2 q^{45} + 11 q^{46} - 6 q^{47} - 2 q^{48} + 7 q^{49} - 2 q^{50} - 2 q^{51} + 3 q^{52} + 4 q^{53} + 2 q^{54} - 2 q^{55} + q^{56} + 3 q^{57} - 4 q^{58} - 14 q^{59} + 2 q^{60} + 9 q^{61} + 5 q^{62} - q^{63} + 2 q^{64} - 3 q^{65} + 2 q^{66} - 23 q^{67} + 2 q^{68} + 11 q^{69} - q^{70} - 2 q^{72} - 16 q^{73} - 15 q^{74} - 2 q^{75} - 3 q^{76} - q^{77} + 3 q^{78} + 2 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} - 13 q^{83} + q^{84} - 2 q^{85} + 2 q^{86} - 4 q^{87} - 2 q^{88} + 28 q^{89} + 2 q^{90} + 19 q^{91} - 11 q^{92} + 5 q^{93} + 6 q^{94} + 3 q^{95} + 2 q^{96} + 13 q^{97} - 7 q^{98} + 2 q^{99}+O(q^{100}) \) 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
3.70156
−2.70156
−1.00000 −1.00000 1.00000 −1.00000 1.00000 −3.70156 −1.00000 1.00000 1.00000
1.2 −1.00000 −1.00000 1.00000 −1.00000 1.00000 2.70156 −1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)
\(11\) \( -1 \)
\(17\) \( -1 \)

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 5610.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5610.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5610))\):

\( T_{7}^{2} + T_{7} - 10 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} - 8 \) Copy content Toggle raw display
\( T_{19}^{2} + 3T_{19} - 8 \) Copy content Toggle raw display
\( T_{23}^{2} + 11T_{23} + 20 \) 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
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 8 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 11T + 20 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 15T + 46 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 32 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 40 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 32 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} - 9T + 10 \) Copy content Toggle raw display
$67$ \( T^{2} + 23T + 122 \) Copy content Toggle raw display
$71$ \( T^{2} - 164 \) Copy content Toggle raw display
$73$ \( (T + 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 40 \) Copy content Toggle raw display
$83$ \( T^{2} + 13T + 32 \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 13T + 32 \) Copy content Toggle raw display
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